In order to use the I2C driver we need to include the apropiate header file. For example in the case of LPC1114 this is i2c_11xx.h.
Once that is completed, we need to set the GPIO pins to alternate I2C function, then reset the I2C peripheral, initialize it and set the clock speed:

In case of LPC1114 the I2C is connected to P0.4 and P0.5:

Chip_IOCON_PinMuxSet(LPC_IOCON, IOCON_PIO0_4, IOCON_FUNC1 | IOCON_DIGMODE_EN );
Chip_IOCON_PinMuxSet(LPC_IOCON, IOCON_PIO0_5, IOCON_FUNC1 | IOCON_DIGMODE_EN ); 
Chip_SYSCTL_PeriphReset(RESET_I2C0); Chip_I2C_Init(I2C0); Chip_I2C_SetClockRate(I2C0, SPEED_400KHZ);

And finally we need to provide a state handling function by defining I2C_IRQHandler() function, set the master event handler and enable I2C interrupts:

void I2C_IRQHandler(void) {
  if (Chip_I2C_IsMasterActive(I2C0)) {
    Chip_I2C_MasterStateHandler(I2C0);
  }
  else {
    Chip_I2C_SlaveStateHandler(I2C0);
  }
}

Chip_I2C_SetMasterEventHandler(I2C0, Chip_I2C_EventHandler);
NVIC_ClearPendingIRQ(I2C0_IRQn);
NVIC_EnableIRQ(I2C0_IRQn);

For sending and receiving data, first we define some buffers, to work with and a I2C transfer structure:

static I2C_XFER_T xfer;

/* Transfer and Receive buffers */
static uint8_t tx[10], rx[10];

When sending data, we fill the structure variable with appropiate values by setting slave address, transfer buffer and transmit transfer size then call Chip_I2C_MasterSend():

xfer.slaveAddr = slave-addr;
tx[0] = byte1;
tx[1] = byte2;
xfer.txBuff = &tx[0];
xfer.txSz = 2;
Chip_I2C_MasterSend(I2C0, xfer.slaveAddr, xfer.txBuff, xfer.txSz);
// i2c bytes sent

On receiving data, we do mostly the same thing, except we fill the receive buffer instead of transmit one, and the receive transfer size, and call Chip_I2C_MasterRead():

// then read it's value
xfer.slaveAddr = slave-addr;
xfer.rxBuff = &rx[0];
xfer.rxSz = 8;			// we read 8 bytes
Chip_I2C_MasterRead(I2C0, xfer.slaveAddr, xfer.rxBuff, xfer.rxSz);

These are compiled from the provided sample file, which seems a bit too complicated for quickly figuring out how one should use this driver.