I had an MRI of my right hip with arthrogram to diagnose labral tear and radiologist said dye went to lymph nodes quickly. Why would this happen?

Inflammation. Assuming that the arthrogram was done correctly, and contrast placed into the joint space, then heart rate and motion will not affect uptake. The contrast placed into the joint space gets absorbed by turnover of the joint fluid with uptake into the lymphatic system, and then flows into the blood vessels. Contrast outside of the joint gets absorbed directly by the blood vessels, and usually not...
Location. Most likely due to the location of the injection (especially if you have active inflammation nearby). Also, moving a lot or having a high heart rate can increase absorption.
Outside joint. When you are doing the injection, to a certain degree you are guessing where the joint is based on the image. It is not uncommon to inject outside the joint initially. In that case, some of the agent may get taken up by the lymph system and go to nodes. It should not be a problem.