Short answer: blame the Romans.

Longer answer: cultures borrow useful stuff from each other. The Greeks borrowed and adapted their writing system from the Phoenicians. Then the Etruscans borrowed the Greek system, and adapted it for their own language. Then the Romans borrowed it from the Etruscans (sometime around the 5th century BC), and tweaked it a little more. Then they happened to conquer most of Europe over the next few centuries, spreading their writing system across the continent while they were at it.

As the Roman empire spread across Europe, dialectal differences started to appear in different regions. Over time, these differences got stronger, and many of those dialects evolved into modern languages like French, Spanish, Italian, Portuguese, Romanian... so those languages use more or less the same alphabet, because they came from a common source.

But the story gets a little more complicated. There are plenty of languages in Europe that aren't descended from Latin, but still use the Latin alphabet. In the case of the Germanic languages (a family which includes modern German, English, Swedish, Norwegian, Icelandic, Danish, Dutch, Swiss, and Belgian), it's more or less because of the Catholic Church. The Germanic tribes originally used the runic script[1], but very few people were literate at the time. As the tribes settled down, they started to need more and more things written down--stuff like histories, or contracts between lords and lieges, or receipts for merchants. The church had lots of bookish people--almost all the bookish people--but they used Latin. So the Latin language and the Latin script became the norm for all kinds of documents, and eventually replaced the Runic alphabets.

Other languages, like Basque or various Gaelics, are a whole other story, but it still boils down to "the church had all the literate people and we had to record stuff, so we did it their way".

I should mention that many languages outside of Europe also share a writing system, or use writing systems that have been heavily influenced by each other. The Devanagari script, for example, is actually used for more languages than the Latin script ever has been. But that's outside my area of expertise, so I can't really comment on that.

[1] Runic systems also evolved at least partially from Latin, though. But an older version of the Latin system than the one kicking around when the Germanic tribes moved into Roman territory. So that's fun and messy.