(someone) claims to (do/be something)

Use "claim to ___" when you want to tell what a person said, but you also want to suggest that you don't believe them, or don't know if what they said is true. For example:

I once knew a guy who claimed to be able to remember every telephone number he'd ever called.

There are two different patterns that people use "claim" in. One is "claim that ___", and the other is "claim to ___":

They claimed to be police officers, so I let them in.

They claimed that they were police officers, so I let them in.

to have (done/been)

Phrases that end in "to" like "claim to ___" have a little different structure from normal sentences:

He is rich. -> He claims to be rich.

He knows Arnold. -> He claims to know Arnold.

He's eaten there. -> He claims to have eaten there.

He's been to China -> He claims to have been to China.

(someone) was abducted by aliens

To "abduct" someone means to "steal" a person. Another word for this is "kidnapping" someone.

There's a well-known rumor that aliens sometimes take people out of their homes or cars into the aliens' spaceship. In the spaceship, they do scientific experiments on the people. This situation is called an "alien abduction". You almost always use the word "abduct" instead of "kidnap" when you're talking about aliens. It's also common when you're talking about people being taken during a war:

The two American journalists who were abducted in Gaza were freed two weeks later.

It's common to find "abduct" in its passive form ("___ was abducted").