The rhythm, or meter, of this poem is iambic pentameter. Iambic pentameter is the most common meter in British poetry (especially pre-twentieth century), and it also the meter that most resembles natural speech in English.
"Iambic pentameter" describes a line that is made up of five ("pentameter") iambs. An iamb is a grouping of two syllables in which the first syllable is unstressed and the second syllable is stressed. A word in English that exemplifies iambic rhythm is "about" (among many others), because we put the greater emphasis on the second syllable. So, in this first line by Sidney, we see the following rhythms (with the stressed syllable denoted by the bold font):
My true love hath my heart and I have his
In addition to sounding the line out and counting the stresses, a good clue to figuring out the poem is written in iambic pentameter is to look at the form. This poem is a sonnet, as are all of Sidney's other poems (Sidney's most famous work is his sonnet cycle Astrophil and Stella). Sonnets in English are always written in iambic pentameter.