Learn Hardcore Serbian: Sećam se tog pogleda.

Because the verb is sećati se, which is a reflexive verb meaning to remember. The little word se is part of the verb and normally has to be there.

So:

sećati se = to remember
sećam se = I remember

Without se, it would sound incomplete or simply wrong in standard Serbian.

What form is sećam?

Sećam is the 1st person singular present tense form of sećati se.

That means it corresponds to I remember.

A few present-tense forms are:

ja sećam se = I remember
ti sećaš se = you remember
on/ona seća se = he/she remembers
mi sećamo se = we remember

In real usage, the subject pronoun is often omitted, so Sećam se by itself already means I remember.

Why doesn’t the sentence include ja?

Because Serbian usually leaves out subject pronouns when the verb form already makes the subject clear.

Here, sećam already tells you the subject is I, so ja is unnecessary unless you want emphasis.

Compare:

Sećam se tog pogleda. = I remember that look.