The short version:
It sounds like you are still using the halogen transformer on an LED bulb? That's a problem, even though the bulbs are the same form factor. From the product description of what you linked: "Do not use halogen transformer, Please use the appropriate LED transformer."
I suspect you either need an LED transformer, or need to go back to Halogen bulbs in the fixture you have to keep the transformer matched to the bulbs.
The technical version, to the best of my knowledge:
The general issue here is difference in resistance.
Ohm's Law: V=IR. V is constant (it's based on how the transformer was designed), R changed. Thus I, the current, changes when you put in the LEDs.
The problem is that the main transformer driving the halogen bulbs is fed from an oscillator (may be a derivative of a half-bridge circuit, though there are other options). When you change loads/currents the oscillation changes. Sometimes that change can be catastrophic (meaning in high voltage situations you start a fire)
Most modern electronic halogen transformers are built to detect a short circuit and alter the oscillation of the circuit to discharge the power in a safe way to avoid a fire. This will dramatically decrease the amount of power being provided to the bulbs (due to the duty cycle being dramatically reduced)... and thus cause them to be very dim.
It is likely that the difference in resistance from the Halogen to the LED bulbs is triggering that protection circuit (or alternatively, the extra components in the protection circuit drawing different current than it was designed to draw is preventing one of the main power transistors in the oscillator from activating correctly in the first place). Either way, you are drawing power through the halogen transformer in a way it was not designed, which - over time - will either kill the transformer or your LED bulbs.