The expectation value is like an average number. In quantum mechanics, it's kind of like what the average of a number would be if you asked a lot of people what number they thought it was. For example, say you had a wheel with numbers 1-10 on it. If you spun the wheel and asked 100 people what number they thought it was, you'd add up all the numbers they said and divide by 100. That number would be the expectation value. In quantum mechanics, the same thing happens, but with particles instead of people. Scientists measure the particles in the system a lot of times and from that data, calculate the expectation value.