What does the -mate suffix mean and where does it come from?

The suffix you are looking for is actually -ate:

word-forming element used in forming nouns from Latin words ending in -atus, -atum (such as estate, primate, senate). Those that came to English via Old and Middle French often arrived with -at, but an -e was added after c. 1400 to indicate the long vowel. The suffix also can mark adjectives formed from Latin past participles in -atus, -ata (such as desolate, moderate, separate); again, they often were adopted in Middle English as -at, with an -e appended after c. 1400.

decimate (v.)

c. 1600, "to select by lot and put to death every tenth man," from Latin decimatus.”

Etymonline