English dictionaries translate ばかり as "just (finished, etc.)" but that doesn't seem to really work because "I just came" is past tense and "私は来たばかりだ" is present tense. ばかり is an adverbial particle in the same vein as だけ and まで, but I don't see how ばかり modifies だ. An entry in a Japanese dictionary describes it as 動作が完了してまもない状態にある意を表す, but that doesn't really tell what the meaning of だ is in those sentences since the meanings of ある and だ are quite different. Is だ in たばかりだ sentences being used the same way as in 私は先生だ (noun) or 私は静かだ (adjective)? Since ばかり is an adverbial particle and not something that can connect だ with a noun, I am confused.
ばかり is one of those words that doesn't work very well as an English part of speech. A translation of these phrases gets the meaning across, but it does not represent the structure of the Japanese at all. But if it helps, think of it this way: saying "来たばかりだ" means "the current situation is that I have just arrived." If you said "来たばかりだった," that would mean "a situation in the past was that I had just arrived"; the meaning becomes pluperfect since you're talking about something that had already happened at an earlier point in time.
For a more rigorous explanation, let's break down the sentence: 私は来たばかりだ into parts.
私は: I-topic/as for me
来たばかり: came-just (just came)
だ: copula/it is (that)
So if you wanted to go for a really liberal translation, you could argue it says, "it is that I just came," where "it" refers to the general situation or something similarly ambiguous. I feel like I could have explained this better, but I hope that helps.