Rita Vrataski (Emily's character) and Cage both state the same thing. They simply felt it gone. Some senses, feelings, or states can be felt intuitively. Like being drunk or a hang over, you just know when it's gone. The Alpha blood must impart a feeling like that as part of it's power. As Full Metal explains to Cage thinly rule:
Rita: Now listen carefully. This is a very important rule. This is the only rule. You get injured on the field, you better make sure you die.
Rita: Last time I was in combat...
I was hit.
I was bleeding out.
Just not fast enough.
I woke up in a field hospital... with three pints of someone else's blood and I was out.
I lost the power. Do you understand?
After the crash, when Cage is tied down, and Rita, having escaped in 3 minutes flat, is about to kill him to trigger a reset:
Rita: Sorry, Cage. [Pulls out Gun to Shoot Cage]
Cage: No! Whoa. Wait! Wait!
I'm out. I'm out. They gave me blood.
I lost the power. I feel it. I can't reset the day anymore.
Considering Rita does not argue with him at all, immediately starts planning their next steps, it's obvious that she knows what he means. She simply felt the power was gone.
Remember, the power comes with a mental component, psychically linking the Alpha (Rita or Cage) to the Omega. It's this intense feeling that is gone that lets them know it happened. Rita, as smart as she is, quickly surmised that her blood loss, like the dead Alpha's blood loss, leads to loosing the power.
In the comic that the movie is based on, "All you need is Kill", Rita and Kenji (Cage's counterpart) still retain the power until the end. They do not lose it by blood loss/transfusion, and the ending is completely different.