Learn Swedish: Borgmästaren lär vara mycket sjuk. - The mayor is said to be very sick.

Lär is a special modal verb used to express hearsay ('I heard that...') or high probability ('It is likely that...'). In this sentence, it shows that the speaker doesn't know for a fact that the mayor is sick, but has heard the rumor from others.

Why is there no att (to) before vara?

Because lär functions as a modal verb here, just like kan (can) or måste (must). Modal verbs are followed directly by the infinitive form of the main verb, so we just use vara without an att in between.

Can I use lär in the past tense to say 'The mayor was said to be sick'?

No, when used to mean 'is said to' or 'is probably', lär is a defective verb. In modern Swedish, it is almost exclusively used in the present tense. To express this in the past, you would have to rephrase, for example using Det sades att... (It was said that...).

What are the parts that make up the word borgmästaren?

It is a compound word made of borg (castle or fortress) and mästare (master). The -n at the very end is the definite suffix, making it 'the mayor' instead of just 'a mayor'.