You saw 締める, which means "to tighten." So when things tighten, and then they get cut off, it's when you've reached the deadline. You know that feeling? When you get close, you feel a tightening in your stomach. The deadline is near! But, then when your tightening gets cut off completely, it's because you've reached the end. The deadline.
The reading is 締める and 切る put together. Notice how the hiragana bits get pushed into the kanji though, that's the interesting part, but also the part that's hard to remember. It's probably because this is a short version of 締め切り, but you'll see 締切 more often.
Did you have any luck with putting the deadline off till tomorrow?