Without going into the detailed events of the Apostolic age, Ante-Nicene period and the following history…
How Christianity spread in the early days is fairly simple and similar to how the world's all faiths grow.
(Courtesy: Washington Examiner)
A group of believers exists. They communicate with other non-believers. Consciously and unconsciously, they get these people to connect with their faith. This is how an idea or belief system spreads -- 10 people turn into 100, then 10,000 and then 1,000,000 and so on.
With Christianity, it all started with Jesus and then his twelve apostles.
At the time, the rulers in different regions also played a significant role, including Roman Emperor Constantine. If a ruler has embraced a faith, the Kingdom automatically follows his suit.
This is how Christianity spread in the earliest days.
Even till this day, that's how it is propagated. But of course, now it touts massively powerful institutions, unimaginable money power, and corporate and political support, which helps its case. This is very similar to any religion and faith, be it Islam, Hinduism or Scientology.
(Courtesy: Standard Media)
Now on to the second part of your question: why did it spread?
There are a lot of factors why Christianity grew and attracted more and more people by passing days.
Of course, Jesus and his so many stories reached masses. His apostles did a great job in spreading the teachings of Jesus.
On spiritual and religious levels, Christianity promised a lot. It promised liberation from sickness and sufferings. It promised to help people rise from poverty and isolation. It gave life a much bigger and deeper meaning. It made people believe in afterlife, immortality and all the other religiously-noble things that almost all religions promise. This attracted many. They all wanted all these things.
On the other hand, the Roman Empire, too, helped the growth of Christianity… although not intentionally. Under the rule, inequality was at a peak. There were a lot of other social injustices that people hushedly opposed. In Christianity, they found an alternative that promoted equality for all, as well as promised to uproot all the social evils. People found it appealing.
Such factors, along with many other socio-political reasons, played a crucial role in helping Christianity grow across the world.
Coming to the last part of the question: did the Bible ask everyone to convert to Christianity?
There are many layer to the answer of this question.
Understand that no religion starts with a group of people getting together and enthusiastically saying "let's start a religion". It starts with an idea, a belief. When that idea or belief spreads, communities form. And then it is those communities that start treating themselves a part of a collective group that ends up being called a religion. (I know, this is a very broad generalization. However, when looked from the macro lenses, this is exactly how things look!)
(Courtesy: Christian Today)
Now, in the early days, when the believers would work to convert people, they wouldn’t necessarily aim to make "religious conversion" because they don't particularly believe in religion themselves.
They would, in simple words, would propagate the idea of making the transition from sin to the moral-driven character. They would talk about conversion from bad to good. They would talk about the transition from treachery to loyalty.
These are conversions propagated with pure intent. And these are the conversions that the Bible (and even other religious books) talk about.
They are not bad. They don’t ask you to leave your faith and jump the ship. They ask you to convert all your bad into good… regardless of what you put under the "religion" heading on online forms.
Sadly, today, conversion, in the religious linguistic, is influenced by political propaganda. It has gotten a whole different meaning that the Bible never intended for.
Hopefully, this answers all your questions.