Forgive me if this is basic stuff that you already know. If a string snaps, aside from the very rare manufacturing error, it usually means it was tightened way above its intended pitch (like, more than several notes higher). In this case, there's a good chance that although it was saying "go higher," you were already too high. It's as if you're driving from Italy to Germany, and your GPS keeps saying "You're too far south, head north," but everyone around you seems to be speaking Swedish. ;)
Why might the tuner be saying the wrong thing? There are a lot of possibilities.
- You have it on a setting that only detects E, A, D, B, and G, the pitches of guitar strings. This can be a useful setting if you're already pretty close to in tune. But it could help your tuner get confused, especially if you were already much higher than the top E.
- If the tuner can be calibrated, perhaps it's been set to read very low.
- Human error understanding what the tuner's saying: For instance, if you have it set on "chromatic," and are tuning an E string, when you're just a tiny bit high it will read "E" and indicate that you need to go down, but once you get closer to F it will say "F" and that you need to go up.
- Most likely: Meh, tuners are fallible. They're just listening to a waveform and trying to guess the pitch. Sometimes they pick up "overtones," the higher notes "contained" in the main note (the "fundamental), and mistake those for the fundamental. More expensive ones do a better job, and they do a better job the "purer" the signal is (i.e. a tuner pedal connected to a pickup is much more reliable than something that's using an onboard mic to hear your acoustic sound).
What should you do about it?
The goal is to get an inner sense of roughly where each pitch ought to be, so you can treat the tuner more as a tool to help you be more precise, but not as the only source of truth—to continue the GPS metaphor, to get a sense that the fjords and glaciers tip you off that you've missed Germany and should disregard the GPS. Starting with a guitar that's in tune, listen closely to each string, and start training yourself to get a "feel" for roughly what it ought to sound like. Hum their pitches. Listen to the distances between adjacent strings, and get a feel for how much bigger the distance is if you skip a few strings, or compare the top and bottom E strings. Keep doing this over the course of weeks.
If your tuner can play notes, not just react to them, get it to play the sound of each string, and compare that to what the string is actually doing. You want to get to where, if the string is almost in tune, you might need the tuner to get it exact, but in the meantime you can say "Oh, that's way low," or "that's a little low," or "that's pretty close," etc. If your tuner can't play back pitches, learn how to find them on a keyboard, or find an app that can. Use this comparison to get "in the ballpark," then rely on a tuner to get you precise, once you get into the range of a half-step or less away.