I'm having problems to understand the definition of the level of significance α. I thought I knew wh

You are misunderstanding what "cutoff" means in the context of the Neyman-Pearson lemma. It is referring to a cutoff of the likelihood ratio in the test, not as a cutoff of what p-values are small enough to be significant.
A p-value (of some results) represents the probability under the null hypothesis of getting results at least as extreme/unusual as your current results. The idea is that seeing results that are unusual under the null hypothesis should be evidence to reject the null hypothesis. To decide how small a p-value is small enough to be significant, you must set a significance level, say $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$; if your p-value is smaller than α, this indicates that your data is unusual under the null, so you reject the null.
Comparing p-values against α guarantees that your Type I error is $\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}$: if the null is true, the probability that your test is wrong is $\le <!--\; \le \; -->\mathrm{\alpha <!--\; \alpha \; -->}$. More generally, this is how you should be thinking about significance levels: as Type I error (probability that a test rejects the null if null is true). This is important for likelihood ratio tests, where there aren't p-values being computed anywhere.
"Cutoff" in Neyman-Pearson
In this context, you are forming a test based on a statistic called the likelihood ratio $\mathrm{\Lambda <!--\; \Lambda \; -->}(x)=\frac{L_1(\mathrm{\theta <!--\; \theta \; -->}\mid <!--\; \mid \; -->x)}{L_0(\mathrm{\theta <!--\; \theta \; -->}\mid <!--\; \mid \; -->x)}$. (I am following your textbook's convention of putting the null in the denominator.)
The test is of the form
reject $\mathrm{\Lambda <!--\; \Lambda \; -->}(x)<C$
do not reject $\mathrm{\Lambda <!--\; \Lambda \; -->}(x)<C$
for some cutoff C. (I'm ignoring the case where the likelihood ratio equals C for simplicity.) The intuition is that if the data x seem to support the alternative hypothesis, the numerator of the likelihood ratio would be large so we would lean toward rejecting; likewise if the data x seem to support the null, the denominator would be large and we would lean toward not rejecting. This is C is what the "cutoff" is in the Neyman-Pearson example. It is not a cutoff for what p-values are small enough to be significant.
How do we choose the cutoff? If C is large, then you reject less often (leading to small Type I error); if C is small you reject more often (leading to large Type I error). If you have set a significance level, then you must set C large enough to avoid a large Type I error. The example you are reading is solving for the smallest C that keeps the Type I error below α.