Why does the current stay the same in a circuit? I was informed that in a circuit, the current will stay the same, and this is why the lightbulbs will light up (because in order for the current to stay the same, the drift speed of the electrons need to get faster).  However, I do not understand why the current needs to stay the same from point to point.
Why does the current stay the same from point to point in a circuit?
 A: Electrons are present everywhere in an electric circuit. When a potential difference is applied to the circuit, an electric field is set up throughout the circuit, almost with the speed of light. Electrons in every part of the circuit begin to drift under the influence of this electric field and a current begins to flow in the circuit immediately.
You have to note here that if the potential difference you're applying is constant as with a D.C battery the electric field remains constant, and thus the current remains constant.
A: Alright, this can actually be pretty easily explained without too many equations and only a single thing to keep in mind: charge cannot pile up inside a metal.
In other words, electrons won't ever pile up within a wire. If they did, even for a tiny amount of time, then they'd repel each other super strongly due to the $1/r^2$ dependence of the electric force electrons exert on one another, until they were once again pretty evenly spread out throughout the metal.
Now, what does this imply? If the electrons can't pile up, it means that if we were to measure the rate of electrons flowing through any cross-sectional area of a wire or a resistor in a circuit (I'm assuming a series circuit here), it must be the same for all the cross-sectional areas!
But, by the very definition of current (the amount of charge that flows through a cross-sectional area of a metal per second), that means the current must be the same everywhere in the (series) circuit! Otherwise, electrons would pile up.

Now I'll more directly address your confusion: the current through a lightbulb.
 
What happens at the lightbulb?
So first, yes, you're correct on how incandescent lightbulb works. Inside of a glass bulb, there's a really thin tungsten filament, and when a current flows through it, it emits light. I'll explain why:
Tungsten has two properties that make it perfect material for composing the filament of a lightbulb:


*

*It has an extremely high melting point.

*It's pretty conductive. 


Now, just like in the way water in a single pipe with two sections of different radii flows with a faster velocity through the thinner sections of the pipe, if we make the tungsten filament super-duper thin, electrons will need to "flow" (drift) faster through the filament that's thinner than the rest of the circuit in order for there to not be charge buildup anywhere on the circuit. The thinner we make it, the greater the velocity electrons will have when going through it! 
Sidetrack, but this is why there's a bigger voltage drop through thinner resistors. In order for electrons to not build up within the circuit, they must flow faster through the thin parts, and in order for them to flow (drift) faster through the thinner parts, there must be a greater electric field pushing them through the thin resistors. A greater electric field means a greater voltage drop!
Now, since electrons are moving super quickly through the thin tungsten filament, they're also smashing more frequently into the atoms of the tungsten filament as compared to the atoms of other parts of the circuit. This makes the tungsten atoms start vibrating really rapidly, which heats up the tungsten filament to extremely high temperatures (hence why it was important to make sure to use tungsten, or another metal with a high melting point).
Aaaand, when the tungsten gets heated to a hot enough temperature, its atoms start emitting light!! (Aaaand I could go more into why this happens if you want, but I think its out of the scope of this question...)
Hope that helped!
A: The original poster explained unambiguously that the question is about spatial dependence, not temporal, i.e. a partial case of the Kirchhoff’s current law was interested about.
Tiny electrons are not things providing high-level integrity of electric circuits; they are only charge carriers. It is a good metaphor: when you send a mail, you do not care about carriers and their speed; you care about integrity of mail and terms of delivery. For an explanation how this integrity is ensured along a non-branching circuit read Why isn't resistance proportional to distance squared
A: Look into Kirchoff's current law.
It is simply charge conservation. If a current of 2 C/s flows in, then 2 C/s must flow out. Because charge is not accumulated anywhere. Inflow must equal outflow at every single point in a steady circuit. Mathematically:
$$\sum I=0 \quad \Leftrightarrow \quad \sum I_{in}-\sum I_{out}=0$$


*

*When you in a series circuit have only one path, then the whole current has to leave along this path. So the current entering the next point on the path is still the original current. Through the entire circuit, the current is therefore the same at any point.

*In a parallel circuit a path might split into two, so the incoming current can be split and smaller portions take each path - in total the sum must be zero at any point, so now the current along each path is not necessarily the same. This is how current flow can be controlled.

A: The current flows in the wires of a circuit, carried by the movement of electrons. At any particular time, if you measure the current at two different places in the same wire, you will get the same reading. This is Kirchhoff's current law in action: all the current entering a point in a circuit must leave that point. Any point on your wire can be seen as a 'node' with two current paths leaving it.
One way to look at this is to think of the flow of current through a wire similar to water flowing in a pipe. (I actually detest this analogy, but it's simple enough.) If you measure the flow at two different points in the same pipe, the readings will be the same, as long as you are looking at the volume of water passing per unit time, in whatever units you want, cubic meters per second, for example. Unless there's a leak, the water has to go through the pipe. The same goes for the electrons: unless there's a fault or a short, the electrons have to go through the wire.
A: It is suggested in both the conservation of energy and Kirchhoff's law both state that any form of energy, including current, cannot be created or destroyed. If this is the case, then current must remain the same, for if it  increases, electrons will pile up, and if it decreases, there will be large gaps between the different electrons causing the circuit to slow down and eventually the circuit will terminate.
This is proved in the scientific equation $\sum I_{IN} = \sum I_{OUT}$ .
A: This is simply due to charge conservation.
