How can they bring guns into the Matrix?

They are using the Construct program which is not necessarily restricted to their "LAN".

The Construct is a virtual workspace (or "loading program") created by the Resistance to run simulations or upload virtual objects when they hack into the Matrix.

From the same wiki,

Later, in preparation for rescuing Morpheus from agents in a heavily fortified government building, Neo and Trinity enter the Construct to gather many weapons and a bomb for use in their mission.

It's a bit like when you are playing an RPG, you can go into options and choose a different weapon or similar.

The practical answer is the one given by Architect. The system is set up to allow this. The Machines use the One to populate Zion. They provide the technology, the hover ships, built Zion. They allow the red pills to take red pills and live, using Agents as pawns to further this goal. The design the Matrix to allow these little rebellions as a check valve against total system failure. The entire thing is a setup in the first place. The small number of redpills running around killing a small number of blue pills doesn't matter on the long run.

As such, they allow these hacks, the use of constructed weapons, to give the illusion of choice. Sure, the Matrix is huge and there may be exploits, but that the machines allow these huge gaps in their wireless security for so long begs the answer. The Matrix is intentionally a honey pot (a computer security concept of a computer that's used as controlled bait for hackers).

The rebels are able to hack in weapons because they are allowed to do it.

It's bad software design in the Matrix or an exploit. :)

Something like this is actually possible in some games following practices I'd consider bad game design (partially intended and unavoidable though).

I had to immediately think of Neverwinter Nights 2, a RPG with an optional online mode, where you might bring your own character to any online world that allows local characters (i.e. characters not exclusively stored on the server).

Basically, MMORPGs follow the simple rule The client is in the hands of the enemy. As such, they will only allow you to take a specific set of parameters with you when you create your avatar. The client can't tell the server how much damage they do, which skills they've unlocked, etc. For example, you might pick a race, class, skin color, hair style, hair color, etc. Sounds a bit familiar? Kinda what they do when entering the Matrix.

However, NWN2 (and some other games) go a bit further: They allow the client to dictate the server what they're wearing or carrying in their inventory, i.e. uploading this when you establish a connection. Now it's getting interesting. And there are also some parallels to The Matrix once again:

Players indeed setup small maps using the map editor that work pretty much like the armory in the movie. You've got one or several rooms filled with all kind of items, like weapons, clothes, potions, etc.

So what they do is enter such a modified map, equip the stuff, then save and exit, entering the actual multiplayer world.

They do the very same thing in the movie: They enter their construct/armory, equip weapons, and then enter the actual Matrix. Of course, that inventory is not really saved with their "savegame", but that additional data or payload is obviously sent when setting up their connection. How it's actually done isn't revealed, but there are multiple possibilities.

Once again, from a real world example, when opening a web page, your browser submits additional data (e.g. cookies), that the script running on the server will be able to load/interpret. If you've heard of the Shellshock vulnerability earlier this year, this is an attack most likely pretty close to what they do when entering the Matrix:

You send a request to the server together ("I want to enter the Matrix.") with some additional payload that's not used normally (i.e. the weapon data/inventory). Due to some bug or security issue, they then get the server to actually retrieve and use that data ("materializing" their weapons).

Whilst the Simulator programs are part of their LAN, they have been able to smuggle virtual items into the matrix - eg: phones for communication, shades and jackets to look cool, etc.

This ports that they are using to hack into the Matrix has also given them the ability to hack into certain amounts of computational resources from the matrix. From these resources they have built small viral programs that helps them generate and smuggle smaller objects into the Matrix. That's how they get to go with - "Guns, lots of guns."

Like any hacker, the amount of resources they consume is limited, else that viral program will not stay hidden for long. This is why when they needed a chopper, they had to steal it. They couldn't just generate one and fly it over to the building.