Do you remember reflexive verbs? A verb is reflexive when the subject in a sentence performs an action on itself, in other words, when the subject and the object are the same. In Spanish reflexive verbs use reflexive pronouns (me, te,se, nos, etc.), which play the role of direct object in the sentence:
 

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Yo me veo en el espejo.
I look at myself in the mirror.
 
Since they involve a direct object, reflexive verbs are also transitive verbs (verbs that take a direct object). Many transitive verbs can be transformed into reflexive verbs. Peinar (to comb), for example, is a classic example of a transitive verb:

Yo peino a mi bebé

I comb my baby's hair

Caption 21, Lecciones con Carolina - Verbos reflexivos

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that can also be transformed into a reflexive verb, peinarse:

Yo me peino

I comb my hair [literally, "I comb myself"]

Caption 20, Lecciones con Carolina - Verbos reflexivos

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On the other hand, intransitive verbs are action verbs that, unlike transitive verbs, don't take a direct object receiving the action. Examples are llegar (to arrive), estornudar (to sneeze), morir (to die), caer (to fall), etc. Consequently, these verbs can't really be transformed into reflexive verbs. So why do we always hear Spanish speakers using reflexive pronouns with these verbs? For example:
 

Si me caigo, me vuelvo a parar

If I fall, I stand up again

Caption 8, Sondulo - Que te vaya mal

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Obviously, me caigo doesn't mean “I fall myself." It just means "I fall," because the verb caer[se] is part of a group of verbs that use reflexive pronouns but are not reflexive verbs. These verbs are called verbos pronominales, verbs that are typically conjugated using a reflexive pronoun that doesn't have any syntactic function. It's just the way these verbs are typically constructed! Another example is the verb morir (to die). Me muero doesn't mean "I die myself"; it just means "I die." The following example uses it as part of an idiomatic expression:
 

No hablemos más de comida

Let's not talk about food

que me muero de hambre.

since I'm starving hungry [literally, "I'm dying of hunger"].

Captions 40-41, Salvando el planeta Palabra - Llegada

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Now, while reflexive verbs like peinarse always need to be used with reflexive pronouns, verbs like caer (to fall) and matar (to kill) can be used either as pronominales (caerse and morirse), or as simple intransitive verbs (caer, morir), that is, without the reflexive pronouns. Therefore, the following expressions are also correct (though maybe just a little less common in everyday speech):
 
Si caigo, me vuelvo a parar. 
If I fall, I stand up again.
 
No hablemos más de comida que muero de hambre.
Let's not talk about food since I'm starving [literally, "I'm dying of hunger"].
 
The really tricky aspect of reflexive pronouns is how to use them, either with verbos reflexivos like peinarse or verbos pronominales like caerse and morirse. Typically, you will use the pronoun before the verb, for example me caigo (I fall), te peinas (you comb your hair). But how do you use reflexive pronouns in a sentence that uses more than one verb, for example an auxiliary verb such as the verb ir (to go) combined with a verb in the infinitive?
 
Voy a caer
I'm going to fall
 
Juan va a morir
Juan is going to die
 
Well, the rule is simple. You either use the reflexive pronoun right before the auxiliary verb:
 
Me voy a caer 
I'm going to fall
 
Juan se va a morir
Juan is going to die
 
Or you use it after the verb in infinitive as a suffix:
 
Voy a caerme
I'm going to fall
 
Juan va a morirse
Juan is going to die
 
And the same rule applies to reflexive verbs like peinarse:
 
Ella se va a peinar = Ella va a peinarse
She is going to comb her hair
 
In fact, this rule applies to all pronouns, even pronouns that are not reflexive (that are used to substitute the direct object in any given sentence), like lo, la, los, las, and te:
 
Como sandía / La como
I eat watermelon / I eat it
 
Voy a comer sandía
I'm going to eat watermelon
 
Voy a comerla = La voy a comer
I'm going to eat it
 

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Coincidentally, comer (as well as other "ingestion verbs") is an excellent example of a verb that is transitive in nature but that is also used as a pronominal verb with reflexive pronouns. For example, it’s also correct to say voy a comérmela (I’m going to eat it).