Learn Hardcore German: Ich hefte die Einladungskarte an den Kühlschrank, damit sie mich jeden Morgen an die Party erinnert. - I pin the invitation card to the fridge so that it reminds me of the party every morning.

In Ich hefte die Einladungskarte an den Kühlschrank, you are moving the card to the fridge and attaching it there, so it’s direction → an den Kühlschrank (accusative).

If you were just describing where the card already is, you would use dative:

Die Einladungskarte hängt am Kühlschrank.
(am = an + dem, dative, location)

Why is it den Kühlschrank and not der Kühlschrank?

Kühlschrank is masculine: der Kühlschrank in the nominative.

With an + accusative (because of movement), the masculine article changes:

Nominative: der Kühlschrank
Accusative: den Kühlschrank

So an den Kühlschrank is simply an + den (masculine accusative), required here because the card is being moved to the fridge.

What exactly does heften mean, and how common is it?

heften means “to fasten, to attach, to pin/staple something (lightly) to something else.”
In this sentence, it implies attaching the card to the fridge with a magnet, tape, or similar.

In everyday modern German, people more often say:

Ich hänge die Einladungskarte an den Kühlschrank.
Ich klebe die Einladungskarte an den Kühlschrank.

heften is correct and understandable, but a bit less common in casual speech than hängen or kleben.

Why is it Einladungskarte and not just Einladung?

Both are possible, but they focus on slightly different things:

die Einladung = the invitation as an event/act or its content (“you’re invited”)
die Einladungskarte = the physical card or piece of paper

Since you are physically attaching a card to the fridge, Einladungskarte is more precise.
You could say Ich hefte die Einladung an den Kühlschrank, but that sounds a bit more abstract and less clearly like a literal card.

What does damit do here? Is it the same as so that?

Yes, damit introduces a purpose clause and is usually translated as “so that” or “in order that.”

Structure:

Main clause: Ich hefte die Einladungskarte an den Kühlschrank,
Purpose clause: damit sie mich jeden Morgen an die Party erinnert.

So the meaning is: I attach the invitation card to the fridge *so that it reminds me of the party every morning.*

damit is used when the subject of the main clause and the subordinate clause can be different (here: ich vs sie).

Could I use um ... zu instead of damit here?

Not naturally in this exact sentence, because um ... zu requires the same subject in both clauses.

um ... zu works like: Ich mache X, um Y zu tun.
(I do X in order to do Y; same “I” in both parts.)

Here, the subject of the purpose is sie (the card), not ich:

damit sie mich erinnert (so that it reminds me)

If you want to use um ... zu, you would need to make ich the subject of the remembering:

Ich hefte die Einladungskarte an den Kühlschrank, um mich jeden Morgen an die Party zu erinnern.
(I attach the invitation card to the fridge in order to remind myself every morning of the party.)

Both are correct, but the focus shifts:

damit sie mich erinnert → the card does the reminding
um mich zu erinnern → I use the card as a tool to remind myself

In damit sie mich jeden Morgen an die Party erinnert, what does sie refer to?

sie here is a third-person singular feminine pronoun and refers to die Einladungskarte.

German pronouns normally refer back to the last suitable noun with the same gender and number:

die Einladungskarte (feminine, singular) → sie

It does not refer to die Party, even though that is also feminine, because Einladungskarte is more salient and appears immediately before in the main clause.

Why is it mich and not mir after erinnert?

The verb jemanden an etwas erinnern takes a direct object in the accusative:

jemanden (Akk.) an etwas (Akk.) erinnern
= to remind someone of something

So:

sie erinnert mich an die Party
(mich = accusative of ich)

mir would be dative and is used with different patterns, for example with the reflexive:

Ich erinnere mich an die Party. (I remember the party.)
- mich is still accusative here (reflexive)

So with erinnern:

jemanden an etwas erinnern → accusative object
sich an etwas erinnern → reflexive accusative

Why is it an die Party after erinnert?

With erinnern, German uses the fixed pattern:

jemanden an etwas erinnern
(to remind someone of something)

The an here always takes the accusative:

an die Party
an den Termin
an meine Verpflichtungen

So:

sie erinnert mich an die Party = it reminds me of the party

Is the word order sie mich jeden Morgen an die Party erinnert fixed, or can I move things around?

In a subordinate clause, the finite verb goes to the end: erinnert must stay last.

Before the verb, you have some flexibility. The default, neutral order here is:

Subject: sie
Object (person): mich
Time: jeden Morgen
Prepositional object: an die Party
Verb: erinnert

So damit sie mich jeden Morgen an die Party erinnert sounds very natural.

You can move jeden Morgen or an die Party to emphasize them:

damit sie mich an die Party jeden Morgen erinnert (unusual, sounds marked) The closer you stay to the standard order, the more natural it sounds.

Why is jeden Morgen in the accusative case?

Phrases like jeden Morgen, jeden Tag, jedes Jahr are “time expressions” that often appear in accusative form as adverbials of time.

Here, jeden is the accusative singular masculine of jeder:

Nominative: jeder Morgen
Accusative: jeden Morgen

This is a normal pattern with repeated times:

Ich gehe jeden Morgen joggen.
Wir treffen uns jeden Freitag.

Why is the present tense erinnert used instead of a future form?

German very often uses the present tense to talk about the near future when the context makes the time clear.

damit sie mich jeden Morgen an die Party erinnert naturally refers to future mornings, but German doesn’t need a separate future tense here.

A literal future tense (erinnern wird) would sound heavy and less natural:

… damit sie mich jeden Morgen an die Party erinnern wird. → grammatically possible, but stylistically odd.

So present tense with a time expression (jeden Morgen) is the most idiomatic choice.