Why is docking a cruise ship in a busy port so challenging?

Why?
All big vessels are very slow to react to changes in speed, often need 10's of miles to go from cruising speed to no motion left. They are also designed to hold their course steady, which makes them very unresponsive to the rudder, specially on the slow speeds needed for docking.

Harbour approaches always have many big ships and often also a lot of smaller vessels. Busy harbours can look like busy road networks but without nice painted lines to keep streams of traffic apart.
Besides that, mostly the water does not have a lot of depth, often there is only one small channel which is deep enough for the really big ships.
This can start quite far out already, often in delta areas there are only a few approaches to the harbours.
Where the actual harbour is on a shipping canal, like in Amsterdam, there might be a requirement that all ship longer than a certain size or sticking deeper than a certain depth do not use their propeller screw, as it might be too close to the bottom of the water and damage the canal.

Legal requirement.
Many areas have a 'pilot' requirement for all big ship, that means that any ship over a given size need a very highly trained special 'captain' who is brought to the ship while still quite far out of the coast.
This requirement is for the harbour and its approaches, for an inland water way, or whatever is deemed best for the local situation. It used to be done by small boats, first rowing boats, then small boats with a powerful engine. These days some companies start using helicopters as far as I understand.
Regularly visiting big ship, like the UK to the continent ferries often have their regular captains train as pilot, with repeated training and exams to prove they are up to the needed level.

Tug.
Tugs, also called tugboats, are smallish boats with an engine which is far out of scale to the size of the boat. Their design is such that they are nimble, very fast to change course and to drop speed.

When near to a harbour or in a narrow shipping channel tugs work in groups, two for an 'easy' job, up to 5 for a very difficult one, all tugs directed by the pilot, so under the instructions of a single captain.
By applying the lines with which the tug pull the big ship in different positions on the ship, by the angle of the line and as such the power, and by the amount of motion each tug applies, the big ship can be moved every direction needed.
Often one of the tugs seems to work against the others, being back to front from the other tug and the ship they move, this can be to apply breaking motion or it can be driven the same direction as the other but with its propeller working back to front.
Some cruise ships have the additional screws that would allow to dock without tugs, mostly the smaller cruise ships, and those might do without tugs on deeper water harbours, specially those harbours with less other traffic.

Other small boats around the cruise ship.
When the cruise ship is anchored away from the actual quay, there will be small boats moving the passengers to and from the ship to the coast.
In any mooring, there might be a small(ish) boat near to deliver drinking water, the fuel to power the main engines and the fuel for alternative use aboard and consumables like food and toilet paper.
On regular cruise ship mooring quays those things can be brought in land-side as well, but when there are not many cruise ship stopping on a regular scheme, the tenders which serve widely different shipping are often used, with access on the water side.

One more boat you may see stop at the cruise ship after mooring, the pilot boat, which collect the pilot as soon as his job is done to bring him out again as soon as possible.