The verbs sagen, ansagen, and besagen appear similar when written in their infinitive forms, but have quite different meanings. In common English usage, there are a number of examples where all three might be translated with the English verb "to say," although for besagen the English verbs "to state," "to imply," "to mention," or "to mean" are usually more accurate, and for ansagen, "to declare," "to introduce," or "to present" are usually better.
Here is the verb sagen in its present perfect (German Perfekt) tense:
Sie haben mal gesagt, dass sich erfolgreicher Fußball in erster Linie durch Schnelligkeit und Präzision auszeichnet.
You once said that successful soccer is, above all, characterized by velocity and precision.
Captions 22-23, Fußball: Saisonpremiere
The verb ansagen, in its Perfekt tense, is written identically to and should be distinguished from its slang adjectival form angesagt, which means "popular," "hip," or "hot" (the latter two in the English slang sense). Here is the verb ansagen in present perfect tense:
Einige Schüler haben lästigem Kabelsalat den Kampf angesagt.
Some students have declared war on annoying cable clutter.
Caption 8, Erfinder: Erfindermesse in Nürnberg
And lastly, here is an example of besagen in present tense:
Zu wenig Einsatz, wenig überzeugend beim weiblichen Geschlecht, besagt die Studie.
Too little effort, less than convincing for the female sex, says the study.
Caption 25, Balztanz: für Fortgeschrittene
To sum up: the verb sagen is the act of saying; the verb besagen is referring to what is stated, such as in a law, a study, or on a sign; and the verb ansagen is referring to the act of stating, usually in reference to declaration, such as declaring war.
Read these posts about about sagen and besagen, and brush up on the conjugation of sagen with this video on Yabla German. For advanced learners, check out what Friedrich Wilhelm Genthe wrote about sagen, besagen and ansagen in the "Handwörterbuch deutscher Synonyme" way back in 1834!