Reasons why an I2C bus requires a minimum value of capacitance on the clock to function?
I can think of three possible mechanisms that may result in the system starting to work if you touch the SCL line with an oscilloscope probe. All three indicate a fault in the unit.

First, it may be a mechanical issue such as a bad solder. When you poke it with the probe, it may repair the connection and signal starts to flow.

Second, it may be a noise issue; crosstalk, coupled noise from any source, and the application of the probe's capacitance dampens that noise. But that would mean that your I2C bus's pull-up resistors are far too small. They should be something like a few kilo-ohms.

Third, your SCL and SDA signals might be switching at the same time, which is a no-no in I2C bus. You shouldn't change both signal states in the same operation; there should be a time delay between any edge in SDA signal to any edge in SCL signal. If the both switch at the same time, the application of the probe capacitance to SCL may delay the SCL edge enough that the I2C device's input data stabilizes before it sees the edge in the clock signal. And this failure mechanism would indicate an error in your microcontroller code.



Edit: For a  reference, here is a picture of a proper i2c waveform from the datasheet of the microcontroller which the asker is using.



Image source: LPC1769 datasheet page 59