These words are kind of special.
Люди exists only in plural. There is no "одна люда" or something like that. Also, it does not work with 2,3 and 4 in a standard way (i.e. 2/3/4 + Plural Nominative). You cannot say "дві люди". It has to be двоє/троє/четверо людей (Plural Genitive). Двоє is a special form that can be used with animate nouns (not only this one, any noun, e.g. двоє братів). The rest is the same, 5 людей, 6 людей etc (Plural Genitive, as always).
Людина is singular, does not have a "stand-alone" plural. You cannot just say "людини" without a number attached to it, neither "багато людин". In plural it works only with 2, 3 and 4 (дві/три/чотири людини). Does not work with 5, 6 etc. (cannot say 5 людин).
It's weird. These are really special cases. Similarly, дитина and діти.
Oh my GOD it's so complicated... It's just that special words end up like this. It's not like we all really sat around in a circle and made up these rules :')