What exactly is meant when it is said that only you can make yourself happy? From Lifehack:

Thinking that having a significant other will complete you.
If you are not a whole person already, then having a romantic partner will
not make you whole. Plus, it puts a lot of pressure on the other
person to “make you happy.” You need to be happy with yourself with or
without someone. Having a significant other doesn’t make you happier.
Only you can make yourself happy.

Also, on many other self help sites I have noticed similar statements.
I don't understand this. For example, because you are a human, you probably want to have a man or woman in your life who would love you both mentally and physically, so how exactly are you going to make yourself happy if you don't have that significant other or can't get one?
Does that quote mean that you need to forget/suppress your human needs?
It's mostly a fairly vapid truism (see the rhetorical sense). Here's a diagram from a popular theory from a positive psychologist at my doctoral alma mater of [what really makes people happy (or not):

(source: edbatista.com)  (Lyubomirsky, 2008)
Thus the truer truism would be, "You can make yourself happier, to some extent..." but it's tricky enough to justify the existence of very many books, studies, even the entire field of positive-psychology, due to catches like the hedonic treadmill (essentially the process by which our pleasures "get old").

As for the need for others, it's something of a personality variable, or two (McAdams &amp; Constantian, 1983), or maybe three by a different theory's construction (i.e., relatedness, not all three here). That is to say it's not universal, much less invariant. In reading @huseyintugrulbuyukisik's answer, consider the following question: https://cogsci.stackexchange.com/q/169/4086 You'll find an answer from me based on a recent evolutionary psychological review there as well. It largely supports your assumption that people will want a significant other sooner or later, and surely this is true for most people.

However, there are certainly exceptions to the false "rule":


Asexuality may best describe as much as 1.1% of the general population (Bogaert, 2004, 2006).


A great documentary on an asexuality awareness advocate is available on Netflix. He strikes this personality psychologist right here as quite happy, healthy, and generative. He seems to suffer most from the transience of adult relationships where they lack romantic commitment. I would argue that this is our cultural problem, not his personal problem.

Asociality is a commonly familiar amotivational characteristic that relates to several other traits, states, and disorders, including:


Introversion, a "Big Five" trait dating back to Carl Jung
Depression, which is a broader state of amotivation and negative affect
Avoidant, schizoid, and schizotypal personality, social anxiety, and autism spectrum disorders


The generally undesirable tone of these exceptions belies the social nature of humanity, as does reflection on the benefits of family and social engagement for longevity...but as it is also the nature of humanity to differ individually, almost no statement of human nature is absolute. See health-relevant Wikipedia pages on solitude, social isolation, and hikikomori.


Several cultural references may serve to flesh out the variation in human experience of solitude:


The history of hermitude, non-cenobitic (eremitic) monasticism, anchorites, stylites, poustiniks, self-sufficiency enthusiasts, loners, and misanthropes
An investigative report on solitarily confined inmates
A nonfiction book and a movie based on it with a particularly poignant ending (mind the spoilers in the preceding links!)
An excellent, recent album by the Australian band Tame Impala, Lonerism, that partly celebrates self-awareness:


  "This album is like...someone growing up and discovering other people and just realising their place is not involved with the rest of the world kind of thing. Like discovering that you're a loner."[27]..."It's about the persona of someone who is really isolated – but not necessarily deliberately. Most of the songs are really about other people, being amongst other people. It's really just the idea of being someone who doesn't feel part of the rest of the world, someone trying to figure out where their place amongst everyone else is, and having a really confusing time with it and then slowly accepting that it's in their blood just to be a solitary wanderer. It's meant to put it in a positive light but also a negative light, as the album progresses."[10][13]
  Parker revealed that the album title and general themes of isolation and introspection were selected when "We were touring so much and doing all these gigs and going to places with people. It reminded me of when I was a kid trying to be social and trying to be a people person and realising that I'm not. The last years of touring and me being a people person is just something that's not in my blood. All these cool parties just make me realise I'm just a total outcast. I just can't really talk to anyone without feeling stupid." and "sometimes I just want to run home and never come out of my bedroom again".[27]




References


- Bogaert, A. F. (2004). Asexuality: Prevalence and associated factors in a national probability sample. Journal of Sex Research, 41(3), 279–287.
- Bogaert, A. F. (2006). Toward a conceptual understanding of asexuality. Review of General Psychology, 10(3), 241–250.
- Lyubomirsky, S. (2008). The how of happiness: A scientific approach to getting the life you want. Penguin.
- McAdams, D. P., &amp; Constantian, C. A. (1983). Intimacy and affiliation motives in daily living: An experience sampling analysis. Journal of Personality and Social Psychology, 45(4), 851–861.