It doesnt have a clear meaning when you write only "sevdiğini": 

In a sentence it can be more clear:

- I know that you like me. (Beni sevdiğini biliyorum.)

- Beni sevdiğini söyle. (Tell me that you like me)

- 

Thank you but can´t it be broken down?

Yes it can, but for an accurate meaning you need to see the full sentence especially the final verb at the end.

sev + diği + n + i

sev = the base/root part of the verb ´sevmek´ to love

diği = subordinating suffix (only used in past/present tense) and part of relative or adverbial clauses. It can mean ´that, who, which, what, when´ depending on what follows next, and the final verb.

n = possessive marker, indicating 2nd person singular = you

i = accusative suffix added to show it is tied to the final verb.

To use the previous example

Beni sevdiğini biliyorum

I know that you love me.

Maybe I´m wrong but I thought the suffix is not diği but dik + i(n).

http://www.turkishclass.com/turkish_lesson_302 

Thanks a lot!!! makes more sense now!!!!

you are not wrong..

dik + i =>diği

(1st. person) sev.diğ.i.m

(2nd. person) sev.diğ.i.n

(3rd. person) sev.diğ.i

bana sevdiğini söyle - tell me that you love me 

sevdiğin.i (-i is accusative)

beni sevdiğini söyledi - s/he told me that s/he loved me

sevdiği.n.i (-n- is the buffer letter, and -i is accusative)