TMP takes place in 2273, at least six years after the events of "Errand of Mercy" (2267) and other TOS episodes with Klingons.
I guess the best explanation is is that both kinds of Klingons existed in the late 23rd century.
If you personally require an in-universe reason, then the idea of two different co-existing races is as good as any.
However, I personally don't feel an in-universe reason is required (or even helpful), and I'm just as happy telling myself that TOS Klingons were supposed to look exactly like TMP/TNG Klingons...
(i.e., the idea that TOS Klingons were supposed to have ridged foreheads was retconned by TMP).