What your trainer might be trying to say, in a matter of words, is that when you do cardio you only burn fat for the time you're doing it. As soon as you stop, your fat burning stops. When you do weights, your body continues to burn fat when you are resting, making your metabolism much more effective.
He does really know what he's talking about - it IS better to incorporate weights or all your cardio work is not as efficient as it could be. Do a good weights workout twice a week - and find exercises that you like to do (don't forget the machines). I really couldn't stand doing free weights. He might try getting you to do girly exercises like my first trainer did.
Try out ALL the exercise machines and the weights (get demos if you need), and choose the ones you enjoy. Start with exercises that use the big muscle groups first - I'd try a pectoral, a lat, a legs (press or squat) and a calf raise. Choose some areas of your body you'd like to make stronger. Don't forget to make sure your protein intake is around 80-100 grams per day, or else your body will eat your muscle, and not your fat. If you ever feel hungry, you're in the dangerzone.
A fantastic educational muscle book to buy is
Strength Training Anatomy by Frederik Delavier. It's my bible.
I promise, you will never look at yourself and say 'shit I look too big'. You will only get stronger and more confident, and when you're at the strength and tone you'd like to see, you'll only enjoy maintaining it. Promise.