If you vouch for someone, you provide evidence or guarantee something on their behalf. If you vouch for your brother, you're saying he's a stand-up type of guy.
First used in the 14th century, vouch comes from the Latin vocitare, ("to call, call upon, or summon"). Sometimes it can mean offering supporting evidence, as when, for example, you're summoned to court to testify on someone's behalf. If you write a letter of recommendation for a student or co-worker, you vouch for their character and abilities. If you take out a student loan, the government will automatically vouch for you, that is, they'll guarantee the loan will be repaid.