Queen Gertrude asks Rosencrantz and Guildenstern (who are visiting with the King and Queen as the scene opens) to please stay for a while with them. Beginning in line 37 the Queen specifically asks:
And I beseech you instantly to visit
My too much changed son.
Gertrude knows something has been bothering her son. She is depending on these two visiting friends of him to achieve two goals. First, she hopes their presence will cheer him up because he has indeed been grieving. Secondly, she hopes they can help find the root cause of what is exactly bothering him. Then, she expects them to report that cause back to her.