What's the point of a current source?
The current in any 2 terminal device is always the same, there's no way for a 2 terminal device to provide "extra" current. You can't have 0A in and 4A out.
What it does do is force the current to be the given value regardless of the impedance across it. (So therefore you can't put in an open circuit an ideal current source, the voltage would go to infinity.)
So a resistor by itself has no current through it. Hook it across an ideal current source and you get (for your example current source) 1A through it regardless of the value of the resistor.
A current source is a voltage source which - in ideal circumstances - creates as much voltage as needed for the specified current to flow. Think of it as a power supply with an adjustable voltage and a fairy watching the current and adjusting the voltage rapidly to preserve the constant current.
An ideal current source would suppply the required voltage in any case, so one with open terminals would output millions of Volts to create an arc through the air - of course this very rarely happens with real current sources.
A current source is not a single circuit element, but a close enough approximation can be made out of ordinary circuit elements. Such current sources are often used to drive LEDs and lasers where little differences in the voltage would lead to dim or burnt-out LEDs, but a current limiting resistor dissipating half the power would also not be acceptable.
Current sources also can be thought of as an opposite of ideal batteries, a.k.a. voltage sources. An ideal voltage source keeps the voltage constant across its terminals; a current source keeps the current constant across its terminals the current flowing through itself constant.
Voltage sources "like" their terminals open - no current flows then, no energy is output. They "hate" being short-circuited - that results in infinite current and molten wires at the least.
On the contrary, current sources "like" their terminals shorted. Then only a tiny voltage is required to drive the specified current. But they "hate" being left open; they put out enormous voltages to try to push the current through.
A current source regulates the current that flows through itself by adjusting the voltage across itself. Regulating a current may mean creating it if it does not exist (which is why current sources are not meant to be in an open circuit as they will try to force one by increasing their voltage until it is maxed out or it breaks through air). It is normal that the same current goes in and out, since the current goes through it - the source is in series with that branch of the circuit.
Current sources can be used to create a current, or to ensure the current in that branch of the circuit is at a known value. When you think about it, a simple resistor with a voltage source can be a current source - a not particularly efficient or compliant/robust one but it still is. If a current source is used to create a current, this current can be injected in other branches as currents add up at each circuit node.
Voltage is generally used to transport information, but current sources are everywhere. From biasing to driving LEDs.