It's kind of like you're saying say say. This is what you say when you answer or greet someone on the telephone. There's a strange story about this. One reason you say 申し申し is because it's hard for foxes to say. So, you're confirming that the other person isn't a fox. The other explanation is that operators would say this, letting the person know on the other line that they're talking by saying "say say" but in a humble way. Either way, this is a telephone hello, so say it next time you talk to someone on the phone who is Japanese... just to make sure they aren't a fox.

We usually don't use kanji for this, so don't use it when you're writing. But you'll probably run into it in the wild so it's good to know!