When to say "mirar + a" vs. simply "mirar"?

[createdamadman]

In the image below, why is it "mira a la luna" but "mira ese/el pájaro" (without the "a")? I'm also reading a short story with the sentence "Los aztecas se comieron a las serpientes." What is the purpose of the "a" in this sentence?

[AngelicaDeAlquezar]

As I've said somewhere before, there isn't much difference in daily speech between this kind of sentences using "a" and not using the preposition. Commonly, the verbs that imply an idea of perception, like "mirar", "observar", "ver", "oír", "escuchar", etc., are followed by the preposition "a".
In daily speech, the usage is rather idiosyncratic, and it depends on what the speaker feels like saying.

Also, sometimes, "mirar" implies looking in some direction, so the verb is followed by "a", "hacia" or even "para".
- Mirábamos al horizonte. (We looked into the horizon.)
- Juan miró hacia la casa. (Juan looked at the house.)
- Cuando nos regañaron, mirábamos para el suelo. (When we were scolded, we were looking at the floor.)

My advice is don't worry, these sentences in your examples are synonymous in daily speech. If any one "corrects" what you're saying, they'll just be showing their personal preference rather than having a grammar principle to defend.

[Rusty]

In your last sentence, it wouldn't be clear who ate whom without the addition of the 'a'. That 'a' designates the direct object, where it may not be evident.

In Spanish, the sentence structure isn't as strict as it is in English. The Spanish subject may follow the verb. Either the snakes or the Aztecs could have been the food being consumed.

In your image examples, the first two describe looking at an object (non-human). They're looking at something.
In the next two pictures, the woman is directing the man to turn and look at something. The motion involved means that the orientation of the other person needs to be changed in order to look at the object. This is the reason for the 'a'.
(You'll see it written correctly and incorrectly, because the two vowels are subject to assimilation.) In other words, don't be surprised to see 'aser' when 'a hacer' is what was meant.