The continuity equation (also called the conservation of mass equation) is basically a restatement of the fact that the fluids we will be  dealing with are, to a good approximation, incompressible fluids. That is, if you fill a glass to the very top with a fluid, you are not going to be able to get any more fluid into the glass, no matter how hard you push!  (The glass will break before you can squeeze any more fluid into the glass!)

When fluid flows through a pipe or a tube of flow, it cannot vanish, nor can more be created. Thus, whatever fluid flows into one end of the pipe or tube of flow (mass of fluid flow per unit time: kg/s), it must flow out the other end at the same rate. (The mass flow rate of a fluid is defined to be the amount of mass of a liquid flowing through the pipe in a given amount of time.  Units: kg/s   The flow rate or volume flow rate of a fluid in a pipe or tube-of-flow is defined to be the volume of liquid flowing through the pipe in a given amount of time.  Units: m³/s)   That is, the number of kilograms per second (Dm/Dt) flowing into the tube of flow must be the same as that flowing out.

For a pipe with a uniform cross-section (like the pipes we’re used to thinking of), a given volume of the pipe can be written as V = A L, where A is the cross-sectional area of the pipe and L the length of the section of pipe under consideration.  Since density is defined to be mass divided by volume, it follows that m = rV, where r is the density of the fluid in the pipe.    The mass flow rate can thus be expressed as m/t = r V/t = rA  L/t = rAv, where v is the speed of the fluid flow. It therefore follows that, for any two points 1 and 2 within the tube of flow,

r₁A₁v₁ = r₂A₂v₂ .

This is a statement of the conservation of mass within a fluid. This equation is also sometimes called the Equation of Continuity.  In the case of an incompressible fluid (constant density, so r₁ = r₂),  this equation takes on the simpler form

A₁v₁ = A₂v₂ .

This equation tells us that the product of the area and the speed is a constant.  The only way this can be true is if, when the area decreases,  the speed increases.  This is something that you are already familiar with!  For example, if you are holding a garden hose without a nozzle on the end, and the water is flowing out of the end of the  hose, you can see that the flow is a rather slow but steady flow (mass of water per unit time).  However, if you place your thumb over the end of the hose where the water is flowing out and press your thumb down on the opening, the water suddenly sprays out very quickly from the small opening that is left. – That is, a smaller area through which the water can flow means the faster the speed of the fluid flow.

Another example is water flow in a river.  If the river is very wide, then the flow will be relatively slow.  However, if the river narrows  appreciably, then the water speed will greatly increase, resulting in what is known as “rapids”. Again – less area for the fluid to flow through means a faster resulting speed of fluid flow.