Learn Hardcore German: Diese Übung macht meinen Rücken stark.

Because Übung is a feminine noun. In the nominative case the strong declension of dies- for feminine is diese.
• Masculine nominative: dieser Übung (wrong gender)
• Feminine nominative: diese Übung (correct)
• Neuter nominative: dieses Übung (wrong gender)

Why is meinen Rücken in the accusative case?

The verb machen here means “to make” or “to render” someone/something into a new state. German uses a direct object (accusative) to show what is being affected. Since Rücken (“back”) is what’s being made strong, it’s the accusative object.

Why does mein Rücken become meinen Rücken? What’s with the -en ending?

mein is a possessive determiner. In the accusative singular masculine, determiners take an -en ending:
• Nominative: mein Rücken
• Accusative: meinen Rücken
Thus “my back” in the accusative is meinen Rücken.

What is the function of the adjective stark at the end of the sentence?

stark is a predicative adjective that describes the resulting state of the object (Rücken). In German, predicative adjectives normally follow the verb and any objects:
Subject – Verb – Object – Predicative Adjective
Diese Übung – macht – meinen Rücken – stark.

Why is stark placed after meinen Rücken rather than before it?

Because it isn’t an attributive adjective (which would directly modify a noun: ein starker Rücken). It’s a predicative adjective (a complement) and German word order puts complements after the object.

Could I say Diese Übung macht meinen Rücken stärker? What’s the nuance?

Yes. stärker is the comparative form.
• stark = “strong” (absolute result)
• stärker = “stronger” (comparison to a previous state)
So stärker emphasizes an increase in strength compared to before.

Is there a single German verb for “to strengthen” I could use instead?

Absolutely. You can use stärken:
Diese Übung stärkt meinen Rücken.
Here stärken (= “to strengthen”) takes a direct object, so you drop machen and the adjective.

What if I talk about multiple exercises? Do I need to change macht?

Yes. With a plural subject you use machen:
Diese Übungen machen meinen Rücken stark.
Subject-verb agreement requires plural Übungen → machen.