You need to find the documentation for the inverter, but here's my guess: many inverters have a "power save" mode which cycles the 110V power on and off. It does this looking for a load to power. If it doesn't see enough load, it shuts down again for a period of time (a few seconds). It does this to save power when nothing is plugged in. Your phone charger has a very low load, not enough to tell the inverter to stay on. Try plugging something more in (a table lamp would do), then try the phone charger also. It should now stay on.
Of course that isn't a good solution to charging the phone. But most inverters have a setting to turn the power saving feature off as well. Then it will power the outlets regardless of load as long as the inverter is on. The downside is the constant draw on the batteries is higher (this is referred to as "idle current" in the inverter specs.).