What you heard is correct. You need to prune off the flowers of first-year June-bearing strawberries because that will produce healthier plants and increase  production the next year because the plants have had a year to take in nutrients that can be used to grow a healthier root system.

That said, Ohio State University's publication, Midwest Home Fruit Production, also advises renovation of strawberry patches immediately after your first harvest. Renovation removes old or diseased leaves, lets the plants get more sunlight, offers an opportunity to fertilize, puts soil up to the root crown and controls seeds. The chapter on strawberries said renovation needs to happen immediately after harvest and before July 15.

OSU professor Dr. Gary Gao told my Master Gardener class that the biggest weed problem in a strawberry patch is strawberries. Renovation will help you keep on top of that. Start by pulling weeds. There are some herbicides that you could use; that is your prerogative. If you’ve kept up with mechanical weeding herbicides shouldn’t be necessary. Remove old leaves by mowing over the patch on the highest setting. Thin plants by spacing them 4-6 inches apart.

Use the opportunity to feed the plants using a half-pound of 10-10-10 fertilizer per 100 square feet of planting. Put down ½ to 1 inch of soil making sure not to cover the root crowns. Give the plants about 1 inch of water per week if there is no rain. 

The patch may not look too great after that but it will reward you the next season with beautiful plants and good production. However, strawberries are not long-term producers and you can expect to replace the patch every 3-5 years regardless of how well you take care of them in between.

Enjoy!