The usual use case for logistic regression is when your outcome, or dependent variable, is a binary categorical variable. The fact that the integers $0$ and $1$ are associated with the two cases is because the logistic function is mapping the result to probabilities of belonging to the class associated with the integer $1$. If you recall the definition of a binomial random variable, it is the sum of Bernoulli random variables. And Bernoulli random variables associate the value $1$ with a 'success' and the value $0$ with a 'failure.' By designating one of the binary classes as 'success,' that means 'failure' is not belonging to the 'success' class, and by default, belonging to the other class. So the output of logistic regression is the probability of belonging to the class that was arbitrarily associated with 'success,' and thus the integer $1$.
If you were instead to try to apply a standard linear regression to try to predict membership in one of the classes, call it class A - and remember, even though we focus on one class, not being a member of this class means membership in the other, as there are only two classes in the binary case - then you would get results that are most likely nonsensical. Let's say you did simple linear regression on you classification problem, and let's assume you got a slope of $1$ and an intercept of $0$ for the following. Now let's compare two independent variable values : $x = 10$, and $x = 20$. Are you to interpret the results of your regression as $x = 20$ is twice as likely to belong to the class A than $x = 10$? If $x = 40$, is that four times as likely? Asymptotically, as $x \to \infty$ how can you become infinitely more sure that some item belongs to class $A$? Doesn't that just mean the probability of belonging to class A approaches 1? The dependent variable's numerical value doesn't have a sensible interpretation in this case. The other thing to notice is that you can set a threshold for when you decide one class or the other with logistic regression. If the consequences for making a mistake are symmetric, then anything with an output value above $0.5$ gets assigned to class A, and anything below gets assigned to the other class. If the consequences are not symmetric, you could move this threshold value accordingly. How could you sensibly move that threshold value to account for asymmetric penalties when the output values are unbounded? The point of the logistic function is that it is mapping the result to a bounded range which represents the probability of belonging to class A, and again, since there is only one other class for 'not A', the probability of belonging to the other class is just the complementary probability. The choice of assigning $1$ and $0$ is arbitrary - you could switch them, and the interpretation would be the same - the only difference is which class you decided to deem 'success.'
