A satchel is, traditionally, a bag for carrying one's schoolbooks around in. In the past, if a school kid wanted to "suck up" to an older or more popular kid, they would offer to carry that kid's satchel home from school. If the answer was no, it was a huge disappointment and meant that that person didn't want to have anything to do with you. When he/she finally said "Yes", it was a source of pride and delight. It should be noted that this idea is mostly associated with public (fee-paying) schools, which were (in the past, at least) the preserve of the rich and were mainly boys-only. That's exactly the kind of school that people like Boris Johnson went to. I only had time to listen to that brief quote but whoever they're talking about seems to desperately want to be friends with Boris, and therefore hopes Boris will (metaphorically) let him carry his satchel home.