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The title might be confusing, I’m not really sure how to word it...

I’m making a half orc character. It says that he gets ability score increases on Strength and Constitution.

Does that just mean that his ability score in Athletics (determined by Strength) goes up but not his main Strength stat?

Or does it mean that his Strength stat goes up, changing the ability score anyway?

You only have six ability scores -- you have modifiers for each ability score (like strength) and modifiers for each skill (like athletics). When you create a character, you'll choose initial values for each of your six abilities (either by rolling dice, or using point buy, or using the standard array). The race you choose will modify some or all of those values, and as you level up you'll be eventually able to increase those values.

Your score in any one particular ability determines your modifier for that ability (subtract ten, divide by two, round down).

• Example(STR 17): 17-10, divide by two 3.5, modifier = 3

Your modifier for skills are equal to your modifier for the relevant ability, plus your proficiency bonus if you're proficient in it.

So for example, let's say your starting strength was 15 at character creation, plus 2 from half orc. So at level 1 you'd write "17" as your strength and "+3" as your strength modifier. Your proficiency bonus at level 1 is always "+2", so your athletics modifier is either "+3" or "+5", depending on whether you chose athletics as one of your initial skill proficiencies.

Your skill modifiers are constantly "updated" to match the relevant ability modifier, so if later on your strength goes from 17 to 18, those +3 modifiers all become +4.

Your ability scores are what your main stats are called. Numbers for things such as Athletics and Acrobatics are called skills. To reference the Player's Handbook (see page 175) on using ability scores

Each of a creature's abilities has a score. a number that defines the magnitude of that ability. An ability score is not just a measure of innate capabilities, but also encompasses a creature's training and competence in activities related to that ability.

To also reference the PHB (page 176)

Each ability covers a broad range of capabilities, including skills that a character or a monster can be proficient in. A skill represents a specific aspect of an ability score, and an individual's proficiency in a skill demonstrates a focus on that aspect.

When your half-orc description tells you that your ability score in Strength increases, your main stat called "Strength" increases (to a maximum of 20), not your Athletics skill. However, because your Strength is higher, the modifier given to both your Strength and your Athletics rolls increases.

Ssee the table on Basic Rules, p. 9:

$begin{array}{|c|c|}
hline
textbf{Score} & textbf{Modifier} \
hline
1 & -5 \
2–3 & -4 \
4–5 & -3 \
6–7 & -2 \
8–9 & -1 \
10–11 & +0 \
12–13 & +1 \
14–15 & +2 \
16–17 & +3 \
18–19 & +4 \
20–21 & +5 \
vdots & vdots \
hline
end{array}
$

Hope this helps!