Suppose, I classify something as 1 when predicted probability of that event is greater than 0.5 (referred as threshold, henceforth) and 0 when predicted probability of that event is less than 0.5.
What happens when I change the threshold value to say 0.7?
I can think of is that now my model is more likely to classify something as 0. Hence it will decrease the probability of type 1 error [reject $H_0$ when $H_0$ is true ($H_0$ is 0)] and increase the probability of type 2 error.
In addition to @Dave's pointer in his comment, I'd like to add simply that a threshold is a hyperparameter. You pick the one that gives you the best compromise between false positives and false negatives. Usually, you are able to tolerate FPs or FNs better than the other, so the right compromise (and so the right threshold) is application-specific. The ROC curve is useful for picturing what's happening. Every point on the ROC curve corresponds to a different threshold.