Why do bosons tend to occupy the same state?

Suppose you have two distinguishable coins that can either come up heads or tails. Then there are four equally likely possibilities, $$\text{HH}, \text{HT}, \text{TH}, \text{TT}.$$ There is a 50% chance for the two coins to have the same result.

If the coins were fermions and "heads/tails" were quantum modes, the $\text{HH}$ and $\text{TT}$ states wouldn't be allowed, so there is a 0% chance for the two coins to have the same result.

If the coins were bosons, then all states are allowed. But there's a twist: bosons are identical particles. The states $\text{HT}$ and $\text{TH}$ are precisely the same state, namely the one with one particle in each of the two modes. So there are three possibilities, $$\text{two heads}, \text{two tails}, \text{one each}$$ and hence in the microcanonical ensemble (where each distinct quantum state is equally probable) there is a $2/3$ chance of double occupancy, not $1/2$. That's what people mean when they say bosons "clump up", though it's not really a consequence of bosonic statistics, just a consequence of the particles being identical. Whenever a system of bosonic particles is in thermal equilibrium, there exist fewer states with the bosons apart than you would naively expect, if you treated them as distinguishable particles, so you are more likely to see them together.

Anyway, according to many science writers, bosons not only can be in the same state, but they also tend to do that. Why is it like that?

That's simply wrong. It's one of many clichés dear to popular science writers but with little physical content. What's true, as you said, is that bosons can occupy the same state, as opposed to fermions. As to tendency, it has the same half-truth as when one says "a system tends to stay in its ground state".

In fact assume the latter statement is inconditionally true and you have an ensemble of non-interacting bosons. Then each boson will tend to occupy its ground state and - since nothing forbids that - all particles will sit there. In presence of interaction things may go different or not depending on the interaction.

But the real issue is if there is the tendency of a system to stay in its ground state or go into it if initially placed in a state of higher energy. However this is a question not easy to deal with in few words.