The petals fall because the flower is over mature and past the pollination stage.  No fruit will form unless the flower was pollinated by some means other than your efforts.  This sometimes happens.

Petal fall is normal after a flower has matured whether it was pollinated or not.  Tiny fruits may form, but any that were not pollinated soon fall off. Because lemon plants produce many more flowers and tiny fruits than they can support, the majority fall off as part of a natural thinning process.  Sometimes it's even necessary to remove some of the fruits that do not fall to get good sized lemons and maintain a healthy plant.

However, if the flowers fall off as soon as they open, or even before, it could be because the plant is too dry, too wet or otherwise stressed. If possible, put the plant outdoors in summer. A sunny but sheltered location is best. Flowers that form outdoors will be pollinated by insects and fruit that develops will ripen indoors in winter.

We didn't see the photo.  The stems will become more sturdy if the plant gets plenty of light and the proper amount of water and nutrients.  Growing outdoors in summer will hasten the process.