I am learning causative forms.
The teacher allowed me to participate.
What does 先生が参加させてもらいました mean?
参加させてもらう also means "being allowed to participate", but the focus is on the subject (who is being allowed), not so much on the "allower". More literally it would be "receiving the favour of being allowed to participate".
But there's an area of caution here. If the second sentence is indeed supposed to be 先生 が... , then it means "the teacher was allowed to participate".
に to mark the subject from whom you are receiving the favour (term?), but in this case it is omitted (possibly because that subject is vague of irrelevant).
However, my inclination is that the second sentence is really supposed to be 先生 に... , in which case it would mean roughly the same thing as the first sentence: The teacher allowed me to participate. However, as I mentioned, in this case, the focus is more on the act of you receiving permission from the teacher, rather than on the teacher's (kind) act of allowing you to participate.
For more information on the distinctions between
〜（て）もらう (and by extension,
〜（て）いただく, refer to the answers in this post.