Because they were ordered to do so by the Senate. Remember, the Jedi Council learned about the clone army at the same time as the Chancellor and his advisers. The discovery of the clones was announced to the Senate. While the latter was voting on the issue, Yoda went to Kamino for a look at the clones. He then took the clone army to Geonosis to back up the Jedi sent to Geonosis to rescue Obi-Wan and face Dooku and the Separatists.
On discovery of the Clone Army, Obi-wan reported to the Jedi Council, who would meditate on the matter, but also wanted Jango Fett brought to them for questioning. Obi-wan could sense no motive from the Kaminoans so he just relayed their story of army being made for the Republic by a now dead Jedi Master some ten years prior. The oddities were form Fett, so that was where the investigation would lead. This lead ended when Fett lead Obi-wan to Geonosis and the Separatist crisis came to a head upon the discovery of the massive droid armies being raised by Count Dooku. First just the Jedi went in, authorized by the Senate, followed by Yoda taking delivery of the Clone Army and bringing it as backup. Since the Jedi ordered it for the Republic and they Jedi were the only Republic members with any sort of authority present the Clones I think decided the Jedi were their superior officers and the Jedi just...we went with it due to the numbers of blaster bolts going over their heads and their need to bring back Dooku and the likes of Gunray in for trail for crimes against the Republic (the Jedi were attached to the Judicial Department prior to taking over the Clone Army). The beheading of Fett by Windu did cause a problem with them no longer having their witness to interrogate. As the Republic was not prepared for war (officially), the Jedi took the active role as leaders of the Army. The Navy was able to pull officers from various Planetary Defense Forces and the Republic's own small navy to command some of their new fleets of Star Destroyers, but most of the new ships in the Republic's Navy were crewed by clones. The amazing part of the contract for the purchase of the Clone Army was not just the clones, it was the other contracts for weapons for this Army and a Navy to support it. Beyond the large Assault ships seen in Attack of the Clones, their new Star Destroyers were in service within six months of the start of the war (and lot of them), as well ad rapid military construction of medical bases, Starfighters, and of course the rest of the equipment we saw on Geonosis for the first battle. Though that might be normal for when one orders an Army from a race well known for Cloning armies. The buy includes side contracts for weapons development and contractors to fit the new weapons in with the buyer's existing aesthetic. In this case, the Jedi's use of wedges and red.
To put it mildly: you'd think tracing the template of this clone army right to the side of a bloke who's trying to precipitate a civil conflict should have raised a little red flag, no? I mean what are the odds that it's a total coincidence that Dooku just so happened to hire the template for the Republics secret army to kill the main opposition to the military creation act? The galaxy ain't that small.
One thing to consider in all this is the Jedi Council are not the government of the republic. Misgivings or not after the Battle of Geonosis, it's the Senate that decides to deploy the Clone Army in the war from that point on. Besides, the only option other than use them would be for either the Jedi to fight the entire war on their own, or to simply surrender and allow the disintegration of the Republic itself.
With hyperdrives...the galaxy can get incredibly small. The Jedi did see some of those red flags, but the crisis and their blindness caused by the Dark Side, clouded their senses, since the Jedi rely on the Force rather than common sense. "Blind we are", as Yoda says. The Jedi also took an active role in the Clone Army as a means of keeping it close to them. I think their reasoning was that if their clones were to be used against the Republic or Jedi, the Jedi might be able to keep that from happening if they can teach the clone and drive their development. The Jedi were almost correct. Their treatment of the Clones generally was rather good and the clones respected many of the Jedi and would never have killed them without Order 66 being engrained into the clones. Prior to TCW making the biochips a thing, Order 66 was thought of in two ways. One: A preprogrammed hypnotic order than could not be disobeyed (similar to the biochips), or Two: A preset plan that the clones knew about and would carry out as they were designed to do. The later was used a lot prior to TCW, but that set up for a lot of clones to fictionally disobey orders due to their connection with their Jedi Generals. The biochip removes the chances of the clones disobeying orders (unless said chip was removed). The novel of Revenge of the Sith seems to have it as a mix of the two. Order 66 is something deeply embedded into the Clones, so that when Cody gets the order he regrets that it didn't come before he gave Kenobi back his lightsaber. But also it is so natural for the clones to just obey an order, they don't question it, they don't get angry, they show no sign of hostile intent against their Jedi Generals. They just open fire. This defeats the Jedi's danger senses, resulting in many of them being taken by surprise, when normally one shouldn't be able to take a Jedi by surprise.
Interesting (although I've gathered that the "Star Wars: Absolutely Everything You Need To Know" book is proving to be unreliable, at best).
Best bet right now is TCW series for what is still canon about the Clone Army and its origins...as well as what the Jedi know, and what they could have known had events not stopped them.
Interestingly enough, when you watch ROTS, when Ayla Secura is killed, the first inkling she gets something is wrong appears to be, not from the actions of the Clone Troopers, but from a nearby creature. Perhaps the animal was more attuned to The Force than the Jedi so picked up even this low level of intent? (Timestamp 2.30) This would fit with the idea that the Clone Troopers, operating calmly and dispassionately, would give nothing away. Another sign of Palpatine's long-term planning.
From TCW we've gotten to know that the clones are independent enough that if Cody randomly ordered a walker to fire on a Jedi, the gunnery crew is likely to at least question the order if not outright refuse it. Given how quickly and unquestioningly the clones turned, is it fair to assume that they *all* got Palpatine's transmission and not just the commanders as they sequence makes it appear? Does it have to be verbal or do they also need to see that image of him on their visors as well? I mean Sidious can't risk Dooku getting idea and triggering it himself, no?
Yeah, a few contradictions. For example, their description of how Kyber crystals work is the non-canon Legends version, not the canonical version we're getting now. In fact, there's even a disclaimer in the back of the book that it uses a lot of Legends material and Legends materials are not canon. At this point, I hold anything it it says with a grain of salt.