OP: be aware that this subreddit does not permit discussion of current events, which for our purposes is 1995 onwards. Since (according to wiki) Al-Qaeda didn't become active until 1988, that leaves only the first 7 years, which may or may not be useful to you. As such, you might consider x-posting this question to a political, regional, religious or social sciences sub.

Trying to keep this within the proper time frame I can shed some light for you. Al Qaeda translates to "The Foundation" or "The Base". It was founded in Afghanistan towards the end of the mujahideen struggle against the Soviets. It's purpose from the beginning was to coordinate and spread the network of fighters and philosophy of jihad that was developed there to other areas of conflict in the world.

It would use the skills and people from Afghanistan to be the foundation for a broader struggle. The reason they were able to succeed at that was in part due to Osama Bin Laden's leadership and connections.

He was an educated man from a very wealthy and well connected family. He had access to the funds and network that could help move people and material from Afghanistan to other parts of the world. Unlike a lot of local fighters he also had a broad global perspective on what Jihad meant. There were quite a few foreign fighters that had travelled to fight in Afghanistan and we're ready to continue that fight elsewhere, which gave him a very strong pool of personnel to draw from.

After the Soviet withdrawal the Afghan government was weak and friendly to their cause, so Al Qaeda had the room to develop without much scrutiny.

After this came the first Gulf War, Al Qaeda and bin Laden offered there services to the Saudi's but were passed over. Instead the Saudis strengthened their alliance with the U.S. and other western nations. This move was not very popular among certain people and their support turned to Al Qaeda whom they considered a more legitimate force to be involved in the land of the Holiest sites in Islam.

Following this the relationship with the Saudis began to strain, and a new objective was necessary. So Al Qaeda was smart to maintain the momentum of the group and go to Sudan after an Islamist coup occurred there. Doing that gave them credibility as true Global Jihadists, and furthered the reach of the network. It also put them in closer contact with Egyptian Jihadist groups that were beginning to face real pressure from the government and people in Egypt. Many of them were looking for a new place to go, and the Al Qaeda presence in Sudan was a good fit.

That takes us up to around '95, so I'll stop the narrative. In summary though, Al Qaeda started off as a group of battle tested fighters in Afghanistan, who coalesced under the leadership of a man who had the vision and means to widen their impact on the world. They moved quickly to position themselves as the world wide standard for jihadists, and stuck to areas with friendly governments that tolerated and supported them for their base of operations. This gave them room to grow as they absorbed fighters, experts, and leaders from other groups. They understood their mission as being global and made smart moves to both protect it in its infancy and grow its legitimacy and capabilities.