"Biodynamic" or "bio-dynamic" refers to what is commonly called "organic" agriculture, which really means ecologically aware, natural, avoiding artificial inputs and processes (such as artificial fertilizers). So it refers to wine made from grapes that are produced organically, and also by a wine-making process that is "natural" and avoids chemical inputs. "Biodynamic" specifically refers to a form of organic farming with a certain esoteric or even mystical perspective.
https://en.wikipedia.org/wiki/Biodynamic_agriculture

(Exactly what is or is not included in "organic" or "biodynamic" farming is not always as clear as people assume; they are to some extent a marketing terms.)

"Leaning" expresses tendency or inclination; it means that the collection of wines offered in the restaurant tend to be of this type; its wine list "leans towards" biodynamically produced wines. It doesn't mean they all are, necessarily, but suggests that many or most are and that this is the restaurant's policy.

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Note added at 6 hrs (2015-12-30 09:02:49 GMT)
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In response to Linda's point: it should be noted that in several respects "biodynamic" wine is not exactly the same thing as "organic" wine, and in order to be labelled "biodynamic" a wine has to be certified by a particular body. In the UK it is an organisation called Demeter, which issues a quite elaborate set of production standards. So the term "biodynamic" does come with baggage attached, as you might say, and a restaurant that favours specifically biodynamic wine as opposed to organic wine in general could be seen as favouring the specific philosophy behind the biodynamic movement. That should be noted.

So I should have said "wine produced biodynamically" rather than "organically". I was trying to make it easier, at the expense of precision.

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Note added at 7 hrs (2015-12-30 10:11:57 GMT)
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I remain unconvinced that the difference between biodynamic and organic is a radical one. The radical difference is between both of these and traditional non-organic. The following is from a scientific study by Jennifer R. Reeve et al., "Soil and Winegrape Quality in Biodynamically and Organically Managed Vineyards",American Journal of Enology and Viticulture, 56 (4) (2005): 367–376:

"Wines produced from biodynamically grown grapes have received increasing attention. Similar to organic agriculture, biodynamics eliminates synthetic chemical fertilizers and pesticides. The primary difference between the two farming systems is that biodynamics uses a series of soil and plant amendments, called preparations, said to stimulate the soil and enhance plant health and quality of produce. Whether these preparations actually augment soil or winegrape quality is unclear and controversial. [...]
The study consisted of two treatments, biodynamic and organic (the control), each replicated four times in a randomized, complete block design. All management practices were the same in all plots, except for the addition of the preparations to the biodynamic treatment. No differences were found in soil quality in the first six years. Nutrient analyses of leaf tissue, clusters per vine, yield per vine, cluster weight, and berry weight showed no differences. [...]
Biodynamic preparations may affect winegrape canopy and chemistry but were not shown to affect the soil parameters or tissue nutrients measured in this study."
http://www.ajevonline.org/content/56/4/367.abstract

Linda Chalker-Scott, a Washington State University horticulturalist, claims in "The Myth of Biodynamic Agriculture" that what distinguishes "biodynamic" from "organic" is pseudo-scientific:

"Given the thinness of the scientific literature and the lack of clear data supporting biodynamic preparations, it would be wise to discontinue the use of the term “biodynamic” when referring to organic agriculture. I am guessing many academics, both theoretical and applied, have no idea where the roots of biodynamic agriculture lie: the fact that “biodynamic” is used interchangeably with “organic” in the literature seems to support this conclusion. For me and many other agricultural scientists, usage of the term is a red flag that automatically questions the validity of whatever else is being discussed."
http://puyallup.wsu.edu/wp-content/uploads/sites/403/2015/03...

The conclusion seems to be that for practical, though not for marketing, purposes, biodynamic is indeed organic, and that claims that it is much more have not been scientifically substantiated.