Let's say I'm a huckster that plays a game with you where if I roll a single six sided die and it lands on 1 you lose but you win otherwise.
Let's say you have some guarantee that I'm using the same die each time and that each of the rolls are independent. We play the game ten times and 1 is rolled the first 9 out of 10 times, with a 5 being rolled on the 10th throw. Now, you know that there's a common loaded die that can be purchased that has a weight to skew the probabilities and you further know that the loaded die rolls a 1 80% of the time and the remaining 20% spread evenly to the other values (so 4% for every other value).
Given a choice between the loaded die and the fair die, which is more likely?
The first model, call it $\theta_0$ is the fair die. The second model, with the unfair die, call it $\theta_1$.
The probability of the first model ($\theta_0$) is:
$p( 1,1,1,1,1,1,1,1,1, 5 ; \theta_0) = \frac{1}{6}^9 \cdot \frac{1}{6}$
(approximately .00000008269085843959)
The probability of the second model ($\theta_1$) is:
$p( 1,1,1,1,1,1,1,1,1, 5 ; \theta_1) = (0.8)^9 \cdot 0.2 \cdot \frac{1}{5}$
(or .00536870912000000000)
So we write a computer program to iterate through all the "models" to see which is the more likely. In this case, the iteration goes through two models.
The models can be Gaussians, with model parameters the mean and variance, say, or some other distribution with other parameters to choose from.
For some conditions on models and their parameterization, we might even be able to use more intricate methods that use calculus, gradient descent, etc. to find the MLE.
The MLE formalism is trying to say "given the observation, which parameters fit the best". It gets more complicated because we have to talk about which distributions we're allowing (which "model") and how we parameterize them. In the above, the models are simple, just assigning different probabilities to each of the outcomes of the die rolls and we only have a choice of two parameterizations.