There are a number of problems with that circuit, including the fact that it simply will not work.
Pin 10 of the LM3914 does not go to 0V when the top-of-scale LED is illuminated - it only goes to about 2V below the positive rail (the exact voltage depends on the LED colour, and the LED current setting), so there will only be about 1.5V across the relay coil, so the relay coil will not energise when the battery is fully charged. Assuming this problem was fixed, there are more problems.
The circuit only monitors the battery voltage, and battery voltage is not an accurate indicator of when full charge has been reached. Depending on the type of battery, monitoring the charge current might be better.
The circuit monitors the voltage after D1. The voltage drop across D1 will vary slightly depending on the circuit load. So the monitor will be inaccurate. R1 should be connected to the positive input of the circuit, i.e. the anode of D1, not the cathode.
The dot graph display monitors the entire range from 0V to whatever limit voltage is selected, but the battery voltage will typically only vary over the top 20% or so of that range. So only the top two or three LEDs will ever tell you anything useful. The others will never be used. This can be fixed by changing the circuitry around the voltage reference and moving pin 4 from 0V to a higher voltage.
The way the circuit is designed, when the battery reaches the maximum voltage, the relay will be activated, and this will turn the charger OFF. The relay coil current will load the battery significantly (a typical 12V relay coil draws 30 mA), at the time when the charger is also turned off. This will discharge the battery unnecessarily quickly.
The relay should be used so that it's energised when the battery is charging, and drops out to disconnect the charger. That would solve those problems, but it would mean that the relay coil current would be being supplied by the charger, and this might affect its operation. It would be better to use a solid state relay, or a triac with an optocoupler, to control the charger. These need around 10 mA compared to 30 mA for a relay coil.
When the battery voltage reaches the top dot on the LED graph, the charger is turned off, and the battery terminal voltage will "settle", i.e. drop. This settling will be more pronounced if there is a load on the battery (e.g. the 30 mA relay coil current) but it will occur anyway. The LM3914 does not have hysteresis, so the relay will be activated (and the charger will be turned off) as soon as the voltage reaches the top dot. When the battery voltage drops, the LM3914 display will move back to the second-to-top dot on the display, and the charger will be started again. The circuit will oscillate slowly like this, starting and stopping the charger repeatedly, at a rate of perhaps every ten seconds or so!
So this circuit will not work at all, but even if the two major errors are fixed, it won't work very well. Where did you get that diagram from? I would try to find a better design!
BTW to answer your original question. That circuit is intended to be connected in parallel with the battery, which is also connected to the charger. It monitors the voltage across the battery, and (supposedly) detects when the battery is fully charged. It WILL draw some current - from the charger, while the battery is charging, and from the battery, when the charger has been turned off. The voltage that it sees will be higher while the charger is running. When the charger turns off, the terminal voltage will sag, or "settle".
