私の趣味は切手を集めることだ。 

This is a correct sentense.

if you want answers to your question, it is better posting a question to "Answer" section.

「こと」in this sentense makes nown from verb. 「集める」is verb. 「集めること」is "to collect" or "collecting".

「切手を集めること」is "to collect stamps"

「…は…だ」or「…は…です」is "is".

ことだ?

watashino shumi ha kitte wo atsumeru koto da.

私の趣味は切手を集めることだ。

My question is what is the function of ことだ in this sentence? In terms of Japanese sentence structure I understand that it is: my hobby stamp collecting.

"こと”

集める is "collect".

集めること is "collection".

The function of "こと" is nominalization of a word.

”だ”

私の趣味（My hobby) = 切手を集めること（collecting stamp)

"is" in English and "だ” in japanese are a little similar in their meanings.

The function of "だ" is finishing the sentence (and conveying your feelings).

You can also say "です" instead of the words.