$\color{purple}{a+\left(b+c\right)=\left(a+b\right)+c}$

Intuitively, we understand, that it is correct.

Indeed, suppose your friend Ann has 3 apples, Bob has 5 apples and Cliff has 4 apples.

Bob and Cliff give you their apples, then Ann gives you her apples: ${3}+{\left({5}+{4}\right)}={3}+{9}={12}$ $\left({a}+{\left({b}+{c}\right)}\right)$.

But, what if Ann and Bob gave you their apples, and then Cliff gave you his apples?

You will get the same number of apples: ${\left({3}+{5}\right)}+{4}={8}+{4}={12}$ $\left({\left({a}+{b}\right)}+{c}\right)$.

So, regardless of order, you will get all 12 apples of your friends.

Warning. This doesn't work with subtraction.

However, associative property of addition works for negative numbers (in fact, for real numbers) as well.

Example 1. ${5}+{\left({3}+{\left(-{4}\right)}\right)}={\left({5}+{3}\right)}+{\left(-{4}\right)}={4}$.

Example 2. ${\left(-{5.89}\right)}+{\left({2.51}+{\left(-{3.4}\right)}\right)}={\left({\left(-{5.89}\right)}+{2.51}\right)}+{\left(-{3.4}\right)}=-{6.78}$.

Example 3. ${\left(\frac{{5}}{{8}}+{3}\right)}+\sqrt{{{2}}}=\frac{{5}}{{8}}+{\left({3}+\sqrt{{{2}}}\right)}$.