Learn Hardcore German: Diese Tatsache motiviert uns, täglich zu üben. - This fact motivates us to practice daily.

Because motivieren takes a direct object in the accusative: jemanden motivieren.
Uns is the 1st person plural object form (it can be accusative or dative). Here it’s accusative, since it’s the direct object of motiviert.
The subject is Diese Tatsache; if we were the subject, it would be wir.

Why is it "Diese Tatsache" and not "Dieser/Dieses Tatsache"?

Tatsache is feminine. As the subject (nominative singular), the correct demonstrative is diese.
Dieser Tatsache would be dative or genitive; Dieses pairs with neuter (or masculine in some cases), so it doesn’t fit here.

What form is "motiviert" here?

Present tense, 3rd person singular of motivieren.
-ieren verbs are regular: ich motiviere, du motivierst, er/sie/es motiviert.
The past participle is also motiviert (without ge-), but here it’s clearly the finite verb (verb-second position).

Is the comma before "täglich zu üben" required?

It’s optional in this case but recommended.
A comma is mandatory if the infinitive group is introduced by um, ohne, statt, außer, als, or if there’s a correlate (es/das/dazu) or it depends on a noun.
Here it depends on a verb and none of those conditions apply, so both motiviert uns, täglich zu üben and motiviert uns täglich zu üben are acceptable; the comma aids clarity.

How does the "zu + infinitive" work here?

Pattern: jemanden motivieren, etwas zu tun.
The understood subject of the infinitive (wir) is the same as the main clause’s object (uns).
Note that this zu is the infinitive marker (not the preposition). It doesn’t govern the dative.

Where do I put "täglich" in the infinitive? Is "zu täglich üben" possible?

Zu must stand directly before the infinitive verb: zu üben.
Adverbs typically come before the whole zu + infinitive group: täglich zu üben.
Zu täglich üben is wrong; zu üben täglich sounds odd.

Can I say "Diese Tatsache motiviert uns zum täglichen Üben"? Does it mean the same?

Yes. That’s a nominalization: zum = zu dem
- das Üben (capitalized), with täglichen in the dative.
It’s a bit more formal; meaning is essentially the same as … motiviert uns, täglich zu üben.

Can I drop "uns" and say "Diese Tatsache motiviert, täglich zu üben"?

Grammatically possible but elliptical and less natural; it implies a generic “people.”
Prefer including the object (mich/uns/ihn …) or use an impersonal alternative like Das bringt einen dazu, täglich zu üben.

Should it be "um täglich zu üben" instead of just "täglich zu üben"?

No. Um … zu introduces a purpose clause (“in order to”), which doesn’t fit directly after motivieren here.
Use um … zu to give the purpose of an action: Wir üben täglich, um besser zu werden.
With motivieren, use the bare zu-infinitive or add dazu: … motiviert uns, täglich zu üben / … motiviert uns dazu, täglich zu üben.

Does "motiviert" here mean the adjective "motivated"? How do I tell?

Here it’s a verb: in main clauses the conjugated verb stands in second position (V2), as in Diese Tatsache motiviert uns, ….
As an adjective you’d use sein: Wir sind motiviert = “We are motivated.”

Can moving "täglich" change the meaning?

Diese Tatsache motiviert uns, täglich zu üben = we practice daily.
Diese Tatsache motiviert uns täglich, zu üben = it motivates us daily (frequency of the motivating), not necessarily that the practicing is daily. Position matters.

Is "täglich" the same as "jeden Tag"?

Both mean “every day.”
Täglich is a one-word adverb; jeden Tag is a time expression (accusative of time). They’re interchangeable here, with only a slight stylistic difference.

Can I replace "Diese Tatsache" with "Das"?

Yes: Das motiviert uns, täglich zu üben.
Das refers back to something just mentioned and is more idiomatic in speech; Diese Tatsache is more explicit/formal.

Can I front the object: "Uns motiviert diese Tatsache, täglich zu üben"?

Yes. German allows fronting for emphasis: Uns is in first position, and the conjugated verb (motiviert) still stays in second position.
This emphasizes who is being motivated.