Connecting an LCD display to msp430 launchpad

Oh, duh, I didn't read your code carefully -- your send SendLcdByte is wrong:

    void SendLcdByte( unsigned char command_0_or_data_1, unsigned char command)
{
    SendLcdUpperNibble( command_0_or_data_1, command);
    // send the lower nibble next
 // AS WRITTEN:
    P1OUT = (command & 0x0F) | (P1OUT & 0xF0);
 // SHOULD BE:
    P1OUT = ((command & 0x0F) << 4) | (P1OUT & 0xF0);
    Strobe( command_0_or_data_1);
}

If bottom left is base, top collector, and bottom right emitter (as is often the case for SOT23 BJT transistors) then it's 2 NPN transistors forming a Darlington. That Darlington is in common collector.

The Darlington seems to pump current into the battery (without any limiting resistor), and a tiny current flows through the 30\$\Omega\$/2.2M\$\Omega\$ resistors. The battery voltage goes almost completely to SVSIN.

If the battery voltage is low enough (4V - 2 B-E junctions) the 4V will supply a tiny base current to the Darlington through the 2M\$\Omega\$. If the transistors have an \$H_{FE}\$ around 100 this will supply a few mA to the battery. Once charged the base current will drop to almost zero and the current to the battery will drop to trickle charge level.

edit
OK, that was a nice story, with some assumptions, but if the battery is not rechargeable, as TiOLUWA says in a comment below, there's not enough here to know the function of the Darlington. The battery has another function than just being monitored by the SVS, so there must be circuitry connected to SVSIN. I can't tell more without seeing more of the schematic.