The reflexive pronoun indicates an action from one to oneself.

Sometimes it may sound wrong to an English-speaker. But “je me souviens” should be interpreted as “I bring back an old memory to myself”. That explains the reflexive pronoun.

Pronominal verbs are not necessarily strictly reflexives.

Reflexive:

• Je me lave (I wash myself)
• Je me lave les dents (I wash my teeth)

Reciprocal:

• Je me bats (I fight)
• Je me bats avec mes ennemis (I fight against my enemies)

Lexicalized:

• Je me souviens (I remember)
• Je m’ennuie (I’m bored)
• Je m’échappe/je m’enfuis (I escape/I flee)

Autobenefactive (familiar):

• Je me fume une cigarette. (I smoke “myself” a cigarette)

Note that with this latter case, je me mange is possible:

• Je me mange une assiette de riz (Like in Spanish: me como un plato de arroz)

In case you’re familiar with the term "Middle voice", this is what the reflexive pronouns have become in the Romance languages.

In addition to the active voice (where the subject performs the action) and the the passive (where subject is affected by an action performed by something or someone else), some languages mark the in-between cases, when the subject isn’t completely in control of the situation, with a third set of syntactic structures or inflexions. That’s what the reflexive pronouns mark in French.

Latin, like English today, had very prototypical reflexive pronouns, that were used whenever the subject and the (accusative or dative) object referred to the same person. In those uses, the subject was simultaneously agent and patient of the verb.

This provided a pathway to extend the reflexive pronouns, to quote John McWorther’s courses, "as a way of indicating even the slightest degree to which one could conceive of an action as happening to a person rather than being effected by the person on something or someone else"

For example, (and in decreasing order of agentivity of the subject):

Reciprocal verbs, where the subjects are both doing something and being done something to:

Ils se regardaient l’un l’autre (They were looking at each other)

Personal involvement construction, to mark that the subject is benefiting personally from the action:

Je me suis mangé un bon couscous (I ate a tasty couscous – or indeed "I ate myself a tasty couscous")

Ils se sont emparés du ballon (They took hold of the ball)

Movement verbs, where the subject is willing the movement, but also the thing being moved:

Elle s‘est assise à son bureau (She sat at her desk)

L’avion s‘envole à seize heures (The plane takes off at 4PM)

Emotion verbs, where the subject is the one feeling the emotion, but the cause of the feeling is someone or something else:

Il s‘est étonné de ta lettre (He expressed surprise about your letter)

Je me suis énervé sur mes voisins (I got angry at my neighbours)

Complex mental events verb, where the distinction between emotion, perception and cognition is fuzzy, and so is the role of the subject. With se souvenir, for example, the subject is the source of the memory, is affected by it, and by the action of remembering it:

Je me souviens de ma bêtise (I remember my own stupidity)

Je me résous à le terminer (I resolve to finish it)

Je m‘aperçois de sa présence (I notice his/her presence)

Change of states verbs, when something affects the subject without them necessarily wanting to:

Je voulais me réveiller tôt aujourd’hui (I wanted to wake up soon today)

Tu vas t‘enrhumer (You’re going to catch a cold)

La situation se corse (The situation is getting worse)

Agentless passives:

Les déchets, ça se recycle (Trash is recyclable/should be recycled)

La porte s‘ouvre (The door opens)

Les armes se vendent trop facilement aux États-Unis (Weapons are sold too easily in the US)

This variety of use is why the French grammatical tradition prefers to speak about "pronominal verbs" instead of "reflexive verbs", meaning verbs a pronoun must be used with. This name isn’t without problems, not least because pronouns of all kinds are becoming more and more obligatory in a variety of contexts in Contemporary French(*), but learners should learn to abstract the name of the reflexives pronouns from their usage and this is a good way to do it. Another term is "pronominal voice", which I prefer.

(*) In a sentence like "Moi aussi, je m’y connais en informatique", all three bolded pronouns are obligatory for most speakers.

Bibliography:

KEMMER Suzanne; "The Middle Voice"; John Benjamins Publishing, Amsterdam/Philadelphia, 1993

John McWhorter PhD Linguistics (Stanford) expounds this, in The Power of Babel (2003), p. 183

Linguistic Narcissism: Inherent Reflexivity Marking

Another area where a language can take a ball and run with it is
reflexivity. All languages have a way of expressing that an action is
done to oneself, such as English’s I bathe myself. Yet in English, we are
only required to mark this for purposes of explicitness and can quite
often leave reflexivity unmarked with no resultant ambiguity-we can
say I bathe without fearing that someone will wonder, “ Well, actually,
whom do you bathe?” But in many other languages, one must mark
the reflexivity overtly in all relevant cases: in French, one must say Je
me lave—Je lave implies that you bathe someone other than yourself;
in German, you say Ich wasche mich, not just Ich wasche.

p. 184

  Furthermore, in most such languages, the habit has extended to
all actions entailing exertion on oneself, most of which an English
speaker does not even conceive of as reflexive: in French, one does
not just slip, but slips oneself—je me glisse; in Spanish, one does not
just sit down, but sits oneself down—yo me siento; in Russian, one did
not tire, but tired oneself—ja utomilsja. This is where the “ changes
itself ” aspect of such languages comes in. In French, one changes
oneself into a swan rather than just changing into it: Je me transforme
en cygne.
  Yet getting a “feel” for such a language entails wrapping one’s
head around the fact that this fixation on marking any hint of exertion on oneself has spread even into actions exerted within, rather
than on, oneself. In French, then, one does not just faint, but “faints
oneself” (Je m’ évanouis); in Spanish, one does not just feel happy,
but “feels oneself happy” (Yo me siento feliz)’, in German, one does
not just remember, but “ remembers oneself” (Ich erinnere mich)—
similarly. This can even apply to nonsentient objects: in Spanish, if
a window broke, it “broke itself”—after all, it didn’t break something else, and thus Se quebró la ventana “The window broke.”