If you're saying something that's a little hard to believe or sounds like an exaggeration, you can put "I swear..." in front of it. It means "I promise that this is true". For example:
I caught a fish yesterday, and I swear it was as long as my arm!
That aerobics class was brutal! I swear, I've never sweated so much in my life.
One confusing point is that people sometimes say "I swear" even though they really are exaggerating:
I swear, the line at the coffee shop this morning was like an hour long!
English speakers are also likely say "I swear..." when they're angry, to show that they're serious about a threat that they're making:
I swear, if you don't be quiet, I'm going to slap you!